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+{"text":"\\section{Introduction}\n\nBoolean Satisfiability Problem (SAT) is a significant NP-complete problem with numerous practical applications -- product configuration, hardware verification, and software package management, to name a few. There is no single efficient algorithm that solves every SAT problem, but heuristics have been developed that are sufficient for solving many real-life tasks.\n\nNeural networks have demonstrated superior performance on many complex tasks, e.g., image and language processing, protein folding,  and playing board and computer games. So we may expect that, in the long run, neural networks could also replace handcrafted heuristics for SAT problems. \n\nThere are attempts for applying neural networks to solve SAT by integrating them in contemporary solvers or replacing them altogether. \\cite{selsam2018learning} proposes training NeuroSAT architecture with single-bit supervision, predicting whether the SAT instance is satisfiable.\nYet, it does not directly produce variable assignments and requires labels for the training that may be troublesome to obtain. \\cite{amizadeh2018learning} introduces unsupervised loss for training a neural network to solve a related Circuit-SAT problem without requiring labels. They use differentiable constraint relaxation to evaluate the network output and penalise the network for violating constraints. That leads to successful training for Circuit-SAT problems.\n\nUsually, neural networks are trained by minimizing the loss which is obtained at the final layer. We ask whether it is beneficial to let the network know the loss it is currently producing and adapting its output correspondingly. That is not possible with the traditional supervised losses since labels are not available at the inference time. Yet, such an approach is sound for an unsupervised loss and, indeed, provides an excellent basis for designing a neural SAT solver. \n\nIn this paper, we introduce a step-wise recurrent neural SAT solver that at each step comes up with a query of variable assignments, evaluates it with an unsupervised loss, and updates its state based on the evaluation results.  At every step, it outputs a target solution that may contain information from all the queries performed so far. \n\nWe prove that the proposed query mechanism employing unsupervised loss function as query evaluator can provide the neural network with information about (a) satisfiability status of a solution if queried at integer points, (b) reveal the structure of the problem instance if queried at fractional (real) points. That provides rich information about the structure, meaning of the problem, and its solution.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/query_making.pdf}\n    \\caption{The proposed query mechanism works by producing a query, evaluating it using an unsupervised loss function, and passing the resulting value back to the neural network for interpretation. It allows the model to obtain the structure and meaning of the solvable instance and information about the expected model output. The same unsupervised loss can be used for evaluating the query and for training.\n    }\n    \\label{fig:query_overview}\n\\end{figure}\n\nThe proposed architecture is evaluated on several SAT problems, and it outperforms the neural baseline on all of them. We also compare it with classical SAT solvers (Glucose 4 and GSAT) on 3-SAT and SHA-1 preimage attack tasks. Appendix of this paper is available on \\url{https:\/\/github.com\/LUMII-Syslab\/QuerySAT\/blob\/master\/appendix.pdf}.\n\n\\section{Query mechanism}\\label{sec:querysat}\nWe hypothesize that allowing a neural network to make several solution trials at the runtime and getting feedback about them can significantly improve network capabilities in some cases. It is easy to implement almost for any neural network, yet depending on the unsupervised loss function, some networks can be especially suitable for such augmentation.\n\nThe simplest such implementation (depicted in Fig. \\ref{fig:query_overview}) consists of two different neural network layers. The first layer ($NN_A$) is given the current state ($s_{r}$) and it outputs a query ($q$) and some hidden state ($h$). We then use the unsupervised loss function ($loss$) to evaluate the query and obtain evaluation results ($e$). It is important to use a loss function that does not require labels for evaluation, as they may not be available at the test time. The evaluation results ($e$) together with the hidden state ($h$) are then passed as the input for the second layer ($NN_B$) that is responsible for interpreting the evaluated query and producing the next state ($s_{r+1}$) and output logits ($l$).  We can calculate the gradient of the evaluation results ($e$) with respect to query ($q$). The gradient can be optionally added as the third input for the second layer to show the network the direction for decreasing the loss \\cite{andrychowicz2016learning}. The query mechanism is conceptually shown in Fig. \\ref{fig:query_overview} and can be defined as follows:\n\\begin{align*}\n        q, h &= NN_{A}(s_{r}) \\\\\n        e &= loss(q) \\\\\n        s_{r+1}, l &= NN_{B}(h,\\; e,\\; \\nabla_q e) \\\\\n        \\mathcal{L} &= loss(l).\n\\end{align*}\n\nCalculating output logits and loss $\\mathcal{L}$ at every step is not noteworthy per se but merely demonstrates that the same loss function can be used for both -- query evaluation and model training. The training loss in principle could be evaluated only once -- at the network's final layer and can be different from the loss used for the query evaluation. Nevertheless, it is essential not to merge the loss (and logits) used for query with those used for training. Although constructed similarly, their meaning and functioning are different (see Appendix \\ref{apx:query_use}).\n\n\\section{Query mechanism for SAT}\\label{sec:query_mechanism}\n\nWe use the Boolean Satisfiability Problem (SAT) as the testbed for validating the benefits of the query mechanism. To this end, we propose an unsupervised loss function for SAT problems in conjunctive normal form and theoretically show that the query mechanism is indeed beneficial -- it gives the model access to the problem structure and contributes to the performance of the model. \n\n\\subsection{Boolean Satisfiability Problem}\nBoolean Satisfiability Problem (SAT) questions whether there exists an interpretation (True or False assignments to the variables) that satisfies the given Boolean formula. We undertake a related problem -- finding a set of variable assignments that satisfies the given Boolean formula. Throughout the paper, we stick to common practice \\cite{biere2009handbook} and represent SAT formulas solely in a conjunctive normal form (CNF) -- conjunction of one or more clauses where a clause is a disjunction of literals.\n\nAny SAT formula can be naturally represented as a bipartite variables-clauses graph, likewise known as an SAT factor graph \\cite{biere2009handbook}. In such a graph, the edge between variables and clauses graph exists whenever the variable is present in the clause. Two types of edges are used to distinguish a variable from its negation.\nVariables-clauses graph of SAT formula with $n$ variables and $m$ clauses can be represented as a sparse $n \\times m$ adjacency matrix. In order to perform batching, several SAT instances can be placed into a single factor graph yielding a single adjacency matrix for the whole batch.\n\n\\subsection{Unsupervised SAT loss}\\label{sec:sat_loss}\nA common way to train neural networks is by using a supervised loss, such as cross-entropy, which matches the network outputs with the correct labels. But such an approach does not work for training variable assignment for SAT due to several possible satisfying assignments for a single SAT instance. Also, obtaining labels involve SAT solving, which is time-consuming for large instances. Moreover, as we want to integrate the loss function into the neural network as a query mechanism, it has to be differentiable and cannot rely on labels as they are not available at the test time. \n\nTherefore, we design a differentiable unsupervised loss function that directly optimizes towards finding a satisfiable variable assignment of the Boolean formula without knowing a correct solution. To this end, we relax the Boolean domain to continuous variables in the range $[0, 1]$ where 0 corresponds to False and 1 to True. The vector of all variable values in the assignment is denoted by $x$. The value $V_c$ for a clause $c$ is obtained by multiplying together negations of all literals in the clause and negating the result and the loss value $\\mathcal{L_{\\phi}}$ for a formula $\\phi$ by multiplying all the clause values of $\\phi$ together: \n\\begin{align*}\n    V_c(x) &= 1-\\prod_{i \\in c^+}(1-x_i)\\prod_{i \\in c^-}x_i, \\\\\n    \\mathcal{L_{\\phi}}(x) &= \\prod_{c \\in \\phi}V_c(x), \\\\\n\\end{align*}\nwhere $x_i$ is the value of $i$-th variable and $c^+$ gives the set of variables that occur in the clause $c$ in the positive form and $c^-$ in the negated form. \nThe values $V_c(x)$ and $\\mathcal{L_{\\phi}}(x)$ are equal to 1 if and only if $x$ is a satisfying assignment of clause $c$ or formula $\\phi$, respectively, and strictly smaller otherwise. So by maximizing $\\mathcal{L}_{\\phi}(x)$ we can hope to find a satisfying assignment to the variables. But this function is not usable in practice since it often yields zero value in machine precision due to multiplying together many clause values, which are all less than one. Therefore, we use the negative logarithm of $\\mathcal{L}_{\\phi}$ and minimize it:\n\\begin{align*}\n    \\mathcal{L}_{\\phi}^{\\log}(x) &= -\\log(\\mathcal{L}_{\\phi}(x)) = -\\sum_{c \\in \\phi}\\log(V_c(x)).\n\\end{align*}\nTaking the logarithm of the loss does not change its minimum\/maximum structure, so minimizing $\\mathcal{L}^{\\log}_{\\phi}$ is equivalent of maximizing $\\mathcal{L}_{\\phi}$.\n\nA model trained by minimizing $\\mathcal{L}^{\\log}_{\\phi}$ produces \"soft\" variable assignments in the range $[0,1]$. We experimentally observe that when the network can find a solution, it is close to binary and rounding the values to the nearest integer produces excellent results. \n\nOther works \\cite{Ryan2020CLN2INV:, pmlr-v97-fischer19a}, albeit in different contexts, have also proposed relaxing SAT formulas to continuous truth values similar to $\\mathcal{L}_{\\phi}$ loss (one not in the log-space) but faces the same problems as $\\mathcal{L}_{\\phi}$ if used for learning SAT solutions directly. Therefore, log-space formulation $\\mathcal{L}^{\\log}_{\\phi}$ of the loss  trades capabilities of modelling general Boolean formulas for a loss function that gives non-zero loss for large CNF formulas and can be directly applied for learning their solutions. \n\n\\subsection{Power of the query}\nThe proposed loss function, when used in the query mechanism, gives the satisfiability status of a solution if queried at the binary points 0 or 1 and reveals the structure of the formula if queried at the intermediate (real) points. To maximize the amount of information gained by the query and to match the internal structure of Graph Neural Network (GNN), we employ queries that return the loss value $V_c(x)$ for each clause at the query point $x$. The power of such query mechanism is formulated in the Theorem \\ref{thm:int_points} and \\ref{thm:instance_to_be_solved}, respectively. The proofs of theorems are deferred to the Appendix \\ref{apx:int_points} and \\ref{apx:instance_to_be_solved}.\n\n\\begin{restatable}{theorem}{thintpoints}\\label{thm:int_points}\nFor a binary query point $x$ the losses $\\mathcal{L}_{\\phi}(x)$ or $V_c(x)$ are equal to 1 if the formula $\\phi$ or clause $c$ is satisfied and 0 otherwise.\n\\end{restatable}\n\n\\begin{restatable}{theorem}{thsolvedinstances}\\label{thm:instance_to_be_solved}\nA single query that returns the loss $V_c$ for each clause $c$ is sufficient to uniquely identify the SAT formula $\\phi$ to be solved.\n\\end{restatable}\n\nThe proof of theorem \\ref{thm:int_points} follows immediately from the definitions of the losses. The proof of theorem \\ref{thm:instance_to_be_solved} shows how to create a query point $x$ in such a way that the literals making up the clauses can be uniquely decoded solely from the clause losses. The proof assumes that we know the variable and clause count of the formula, but this assumption can be relaxed by padding all the formulas to some fixed maximum size. A stronger result that a single query returning only the total loss $\\mathcal{L}_{\\phi}$ (a single real number) might also be shown by using more intricate reasoning, but we did not pursue this direction since we use per-clause loss. \n\nThe construction used in theorem \\ref{thm:instance_to_be_solved} assumes sufficiently high precision of the numbers, exceeding the limits of standard floating-point arithmetic. Regarding that, in our implementation, we issue several queries in parallel (up to 128) and organize the computation in multiple recurrent steps in which queries are performed repeatedly and one query can depend on the results of the previous one. Such design significantly enhances the power of queries and relieves the need for high precision.\n\n\n\\section{QuerySAT}\nWe validate the theory in practice by building a neural SAT solver that we call QuerySAT and evaluate its performance on a wide range of SAT tasks - k-SAT, 3-SAT, 3-Clique, k-Coloring, and SHA-1 preimage attack. Experimental evaluation is performed on a single machine with 16GB RAM and a single Nvidia T4 GPU (16GB) using AdaBelief optimizer \\cite{zhuang2020adabelief}. Code for reproducing experiments is implemented in TensorFlow and is available at \\url{https:\/\/github.com\/LUMII-Syslab\/QuerySAT}.\n\n\\subsection{Model}\\label{sec:querysat_model}\n\n\\begin{figure*}[ht]\n     \\centering\n     \\begin{minipage}[t]{0.32\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/variable_generalization_3sat.pdf}\n         \\caption{Part of fully solved 3-SAT instances of the test set depending on the variable count. Models were trained with 32 recurrent steps on formulas with up to 100 variables.}\n         \\label{fig:accuracy_on_datasets}\n     \\end{minipage}\n     \\hfill\n     \\begin{minipage}[t]{0.32\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/generalization_steps_3sat.pdf}\n         \\caption{Part of fully solved 3-SAT instances with 400 variables depending on the steps taken at test time. Models were trained with 32 recurrent steps on formulas with up to 100 variables.}\n         \\label{fig:accuracy_on_steps}\n     \\end{minipage}\n     \\hfill\n     \\begin{minipage}[t]{0.32\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/neurosat_vs_querysat_cactus.pdf}\n         \\caption{Time vs fully solved instances trained with 32 recurrent steps on 3-SAT instances with 5-100 variables and then tested on instances with 5-405 variables.}\n         \\label{fig:neural_solver_cactus}\n     \\end{minipage}\n\\end{figure*}\n\n\nQuerySAT is based on a GNN employing the proposed query mechanism and unsupervised SAT loss function. It receives a CNF Boolean formula $\\phi$ in the input represented as two adjacency matrices of variables-clauses graphs - $A_{p} \\in \\{0,1\\}^{n \\times m}$ and  $A_{n} \\in \\{0,1\\}^{n \\times m}$, where $n$ and $m$ are the number of variables and clauses, respectively.  \n$A_{p}$ represents all positive variable mentions in the clauses, and $A_{n}$ represents all negated variable mentions. The network outputs a vector $out \\in [0,1]^n \\subseteq \\mathbb{R}^n$ -- a variable assignment. \nQuerySAT works in a step-wise manner with recurrent application of the following graph-based recurrent unit: \n\\begin{align*}\\label{formula:querysat}\n   \n    q_i &= \\sigma (\\text{MLP}_q(v_i,\\; t)) \\\\\n    e_i &= \\mathcal{V_{\\phi}}(q_i) \\\\\n    c_{i+1} &= \\text{PairNorm}(\\text{MLP}_c(c_i,\\; e_i)) \\\\ \n    v_{i+1} &= \\text{PairNorm}(\\text{MLP}_v(v_i,\\; A_{p} c_{i+1},\\;  A_{n} c_{i+1},\\; \\nabla_{q_i}e_i)) \\\\\n    out &= \\sigma (\\text{MLP}_o(v_{i+1})).\n\\end{align*}\n\nAt the beginning an empty state vector initialized with all ones is allocated for each variable and each clause and the unit is recurrently applied to them $s_{train}$ steps at training and $s_{test}$ steps at evaluation. At each step $i$ starting from $i=0$, QuerySAT produces a query $q_i \\in \\{0,1\\}^{n \\times d}$, where $d$ is the feature map count, by applying a  2-layer multi-layer perceptron $\\text{MLP}_q$ to the variables state $v_i \\in \\mathbb{R}^{n \\times d}$. A random noise vector $t \\in \\mathbb{R}^{n \\times r}$ is also given in the input to this MLP. The query is range-limited by the sigmoid function $\\sigma$ and evaluated by our unsupervised loss. Here we employ per-clause loss defined as $\\mathcal{V}_{\\phi}(q_i)=\\{V_c(q_i))\\; | \\; c \\in \\phi\\}$ which returns the vector of clause losses $e_i$. \n\nWe then obtain the new clauses state $c_{i+1} \\in \\mathbb{R}^{m \\times d}$ by applying a 2-layer perceptron $\\text{MLP}_c$ and PairNorm \\cite{zhao2019pairnorm} to the previous clauses state $c_i$ and the query result $e_i$.  Query mechanism thus replaces a variables-to-clauses message passing step that would be present in a classical message-passing architecture.\nThen, information aggregation from clauses-to-variables is performed by message-passing, which for some variable sums all the clause states in which the variable occurs. Positive and negative occurrences are treated separately and are implemented as a sparse matrix multiplication between the clause state $c_{i+1}$ and occurrence matrices $A_{p}$ and $A_{n}$.\n\nThe new variables state $v_{i+1} \\in \\mathbb{R}^{n \\times d}$ is then obtained  by applying a 3-layer perceptron $\\text{MLP}_v$ and PairNorm to the variables state $v_i$, aggregated clause messages and the query loss gradient with the respect to query $q_i$. The new variables state is then mapped to the output variable assignment $out \\in [0,1]^n$ using another 2-layer perceptron $\\text{MLP}_o$ and sigmoid function $\\sigma$. In all MLP layers we use LeakyReLU activation. \n\nThe model is trained using the loss function $\\mathcal{L}_{\\phi}^{\\log}$ proposed in section \\ref{sec:sat_loss}. The loss is calculated from variable assignments $out$ at each step, and the sum of all losses is minimized. Using the loss at each time-step has shown performance improvements \\cite{palm2017recurrent, amizadeh2019pdp, ozolins2020matrix} versus a single loss calculation at the end. Also, it enables using many more steps in evaluation than in training. Without impacting the model's performance, we use a conditional exit at any step if the variable assignment satisfies the input formula.\n\nAlthough giving an additional noise parameter $t$ to $\\text{MLP}_q$ seems like a minor detail, it serves several purposes. First, it avoids a blowup in the gradient during training, which may result from applying normalization to zero values. Secondly, it enables the model to learn a randomized algorithm by providing greater diversity between the queries, which is especially needed for obtaining good accuracy in testing with many steps. Thirdly, it allows the model to perform differently in several evaluations (we observed that in the experiments), mimicking restarts in the classical SAT solvers. We find that for our purposes $t \\in \\mathcal{N}(0,1)^{n \\times r}$, where $r=4$, works well.\n\n\\subsection{Training tricks}\\label{sec:tricks}\nIn this section, we describe the model architecture and training tricks that are not per se important for the power of the proposed architecture yet slightly improves model performance.\n\n\\textbf{Gradient scaling.} As we calculate loss and minimize it at each step, large gradient values may accumulate in the backwards pass for the first-time steps. Such uneven gradient distribution slows down training, and to mitigate that, we introduce gradient scaling and apply it to the variables and clauses states. It works by downscaling the gradient in the backward pass at each time-step as follows: $stop\\_gradient(x)\\alpha + x(1-\\alpha)$, where $x$ is value and $\\alpha \\in [0,1]$ is a hyperparameter. We find that for QuerySAT with 32 recurrent steps, $\\alpha = 0.2$ works the best.\n\n\\textbf{Multi-assignment loss.} Instead of returning a single variable assignment, we let the model return several $out \\in [0,1]^{n \\times u}$, where $u \\in \\mathbb{Z}^+$ is hyperparameter. The $u=8$ works best for us. For each assignment, we calculate loss value using the proposed loss function $\\mathcal{L}_{\\phi}^{\\log}$ and then obtain the final step loss as a weighted sum of assignment losses. Weighting is done as follows - we sort assignments losses in descending order and enumerate them from $1$ to $u$. Let's call each such number the loss index. Then we calculate the final loss as the sum of each loss value multiplied with its squared index and then dividing the sum with the sum of squared indices. From our observations, such multi-assignment loss only gives improvements when used with unsupervised loss as the model in the early stages of training is free to explore various outputs. The assignment with the lowest loss value (before weighting) is promoted as the final model assignment. \n\n\\subsection{Evaluation}\\label{sec:querysat_evaluation}\n\n\\begin{table*}[ht]\n\\centering\n\\caption{Mean test accuracy (higher is better) as per cent of fully solved instances from the test set over 3 consecutive runs. Both models were trained with 32 recurrent steps for 500k training iterations and then tested with 32, 512, and 4096 recurrent steps. Value after $\\pm$ indicates the standard error.}\n\\label{tab:sat_tasks_results}\n\\begin{adjustbox}{max width=\\textwidth}\n\\begin{tabular}{lcccccc} \n\\toprule\n\\multirow{2}{*}{Task} & \\multicolumn{3}{c}{QuerySAT} & \\multicolumn{3}{c}{NeuroCore}  \\\\ \n\\cmidrule(lr){2-4}\\cmidrule(lr){5-7}\n                                       & $s_{test} = 32$ & $s_{test} = 512 $ & $s_{test}=4096$         & $s_{test} =32$ &$s_{test} = 512$ & $s_{test} = 4096$           \\\\ \n\\toprule\nk-SAT                                  &$72.12 \\pm 0.19$    &$96.61 \\pm 0.78$     &$\\boldsymbol{99.05} \\pm 0.38$            &$21.64 \\pm 0.27$   &$46.85 \\pm 5.02$    &$50.82 \\pm 6.41$              \\\\ \n\\midrule\n3-SAT                                  &$61.89 \\pm 5.19$    &$88.20 \\pm 4.01$     &$\\boldsymbol{93.32} \\pm 3.21$            &$28.38 \\pm 3.24$   &$53.49 \\pm 3.94$     &$57.63 \\pm 4.38$           \\\\ \n\\midrule\n3-Clique                               &$82.00 \\pm 4.73$    &$93.06 \\pm 4.67$     &$\\boldsymbol{94.74} \\pm 4.62$                  &$1.03 \\pm 0.69$    &$1.03 \\pm 0.66$     &$1.04 \\pm 0.66$               \\\\ \n\\midrule\nk-Coloring           &$91.70 \\pm 1.01$    &$97.76 \\pm 0.98$     &$\\boldsymbol{98.32} \\pm 0.82$                  &$0.0 \\pm 0.0$    &$0.0 \\pm 0.0$     &$0.0 \\pm 0.0$               \\\\ \n\\midrule\nSHA-1                  &$33.25 \\pm 4.17$    &$\\boldsymbol{46.57} \\pm 1.16$     &$46.45 \\pm 1.10$                  &$0.00 \\pm 0.0$    &$0.27\\pm 0.09$     &$0.24 \\pm 0.09$            \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{table*}\n\nWe evaluate the QuerySAT model on several SAT tasks, which are standard benchmarks for classical SAT solvers, and the majority of them are not an easy feat for neural solvers. Namely, we chose k-SAT, 3-SAT, and also 3-Clique, k-Coloring, and SHA-1 preimage attack problems represented as CNF Boolean formulas. All datasets consists only of satisfiable formulas. The QuerySAT architecture is compared to the derivative of NeuroSAT \\cite{selsam2018learning} that was used for predicting unsatisfiable cores by \\cite{selsam2019guiding} but is generally applicable to any variables-wise predictions on SAT factor graph. Further on, we refer to this NeuroSAT derivative as NeuroCore. The results are also compared to the GSAT and Glucose 4 classical solvers. For all tasks, we generate a train set of 100k formulas and validation and test sets of 10k formulas each. The 3-SAT and SHA-1 tasks are an exception as their test sets consist of 40k and 5k formulas, respectively. For most tasks, we use larger formulas in the test set to evaluate the generalization capability of the model.\n\nThe k-SAT task is taken from \\cite{selsam2018learning} -- each clause in the $n$ variable formula is generated by sampling a small integer $k$ (size of the clause), and then randomly without replacement taking $k$ of the $n$ variables. Each variable in the clause is then negated with 50\\% probability. The resulting clauses consist of roughly four variables on average. The train and validation sets consist of formulas with 3 to 100 variables, but the test set -- with 3 to 200 variables.\n\nGraph and 3-SAT tasks are generated using the CNFGen library \\cite{lauria2017cnfgen}, which allows encoding several popular problems as SAT instances. We generate hard 3-SAT instances at the satisfiability boundary where the relationship between the number of clauses ($m$) and variables ($n$) is  $m = 4.258n + 58.26n^{-\\frac{2}{3}}$ \\cite{crawford1996experimental}. The train and validation sets of 3-SAT tasks consist of formulas with 5 to 100 variables, but test set -- with 5 to 405 variables. \n\nFor the graph-based tasks, we generate  Erd\u0151s\u2013R\u00e9nyi graphs with edge probability $p$. For the 3-Clique task, where the main goal is to find all triangles in the graph, we use $p=3 ^ \\frac{1}{3} \/ (v(2 - 3v + v^2))^ {\\frac{1}{3}}$, where $v$ is the vertex count in the graph. For the k-Coloring task we use $p = \\frac{(1 + 0.2) \\ln v}{v} + 0.05$ and the goal is to color the graph with at least $k$ colours. Such generation produces mostly sparse connected graphs that are colourable using 3 to 5 colours. The graphs are then encoded as SAT instances using the CNFGen library. For both tasks, train and validation sets consist of graphs with 4 to 40 vertices, but the test set -- with 4 to 100 vertices.\n\nWe also experiment with the SHA-1 preimage attack task from the SAT Race 2019 \\cite{skladanivskyy_minimalistic_2019}. The goal of this task is to find the message value given the hash value. The original SAT competition task uses the SHA-1 algorithm with 17 rounds and asks the solver to find the first up to 160 message bits. Such configuration produces SAT instances at the threshold of satisfiability, and it is expected that only a single solution exists. That is a challenging task even for modern SAT solvers; hence, we use a smaller set-up to find the first 2 to 20 bits of the message. But even then, it is still a moderately hard task. We generate instances for this task using the CGen tool \\cite{skladanivskyy2020tailored}.\n\nTo make the QuerySAT and NeuroCore architectures comparable, we train NeuroCore with the same per-step loss and apply the same bag of tricks that we used for QuerySAT, as described in Sections \\ref{sec:querysat_model} and \\ref{sec:tricks}. NeuroCore is also given a chance to return results in any step if the correct solution has been found. The rest of the architecture is left intact.\nFor both models, we use 128 feature maps that produce similarly sized models and train them with a batch size of 20000 nodes (max node count in the input factor graph), 32 recurrent steps and a learning rate $2 \\times 10^{-4}$. Hyperparameters are selected by performing a grid search by hand. On the SHA-1 preimage attack, models are trained for 1M iterations, but on the rest of the tasks for 500k iterations. Afterwards, models are tested with 32, 512, and 4096 recurrent steps and the mean accuracy as a percentage of fully solved instances over 3 consecutive runs is depicted in Table \\ref{tab:sat_tasks_results}. We see that QuerySAT with 4096 steps performs consistently the best for all the tasks. It can solve 3-Clique, k-Coloring and SHA-1 tasks on which NeuroCore produces an accuracy of almost zero.\n\nDetailed comparison of both architectures is conducted on the 3-SAT task, where we check step-wise and formula-wise generalization of both models. To evaluate both properties, we use models trained with 32 recurrent steps on formulas with 3 to 100 variables. Fig. \\ref{fig:accuracy_on_datasets} depicts generalization to harder formulas (up to 400 variables) with 32 and 4096 recurrent steps at the test time. Fig. \\ref{fig:accuracy_on_steps} shows step-wise generalization by changing the number of steps $s_{test}$ from 4 up to 131k when testing on the same formulas with 400 variables. QuerySAT outperforms NeuroCore on both generalization tasks by a wide margin and, with 32 steps, has a similar performance to the NeuroCore with 4096 steps. QuerySAT's performance increases with the step count at the test time, although it is trained only with 32 steps. NeuroCore, on the contrary, plateaus at approximately 16k steps and 40\\% accuracy. In Fig. \\ref{fig:neural_solver_cactus}, the cactus plot is shown comparing NeuroCore, and QuerySAT tested with 4096 recurrent steps on 3-SAT instances with 5 to 405 variables. Cactus plot shows cumulative time spend for solving instances (y-axis) versus cumulative solved instance count (x-axis). More solved instances in less time indicate better model performance. At the same time interval, QuerySAT solves $\\sim 36k$ formulas, while NeuroCore solves only $\\sim 18k$ from the total of 40k 3-SAT formulas in the test set (see Fig. \\ref{fig:neural_solver_cactus}).\n\n\\begin{figure*}[ht]\n     \\centering\n     \\begin{minipage}[t]{0.48\\textwidth}\n         \\centering\n         \\includegraphics[width=0.77\\textwidth]{figures\/sha1_querysat_vs_gsat_vs_glucose.pdf}\n         \\caption{Cactus plot representing the number of solved SHA-1 preimage attack instances vs. used time for QuerySAT, GSAT and Glucose 4 solvers.}\n         \\label{fig:sha1_query_vs_gsat}\n     \\end{minipage}\n     \\hfill\n     \\begin{minipage}[t]{0.48\\textwidth}\n         \\centering\n         \\includegraphics[width=0.77\\textwidth]{figures\/3sat_querysat_vs_classical.pdf}\n         \\caption{Cactus plot representing the number of solved 3-SAT instances vs. used time for QuerySAT, GSAT and Glucose 4 solvers.}\n         \\label{fig:scatter_query_vs_gsat}\n     \\end{minipage}\n     \\begin{minipage}[t]{0.48\\textwidth}\n         \\centering\n         \\includegraphics[width=0.77\\textwidth]{figures\/training_k_sat.pdf}\n         \\caption{Error as per cent of unsolved instances in the validation set depending on training iteration for the k-SAT task. The validation set consists of 10k formulas with 3-100 variables. $s_{test}=64$ is used for validation.}\n         \\label{fig:training_k-sat}\n     \\end{minipage}\n     \\hfill\n     \\begin{minipage}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=0.77\\textwidth]{figures\/training_clique.pdf}\n        \\caption{Error as per cent of unsolved instances in the validation set depending on training iteration for the 3-Clique task. The validation set consists of 10k graphs with 4-20 vertices. $s_{test}=64$ is used for validation.}\n        \\label{fig:training_k-clique}\n     \\end{minipage}\n\\end{figure*}\n\nTo show current capabilities of QuerySAT, we also compare it to GSAT \\cite{selman1992a} and Glucose 4 \\cite{audemard2018glucose,een2003extensible} classical solvers. GSAT is a widely known incomplete local-search solver, and Glucose is a contemporary conflict-driven clause learning solver. As the QuerySAT is also an incomplete solver, it's directly comparable to the GSAT algorithm. Comparison to Glucose is not on equal ground since Glucose can certify that some instance is UNSAT while QuerySAT runs indefinitely for such instance. That said, we present their comparison nonetheless to give a rough idea of their relative performance. Note that both - GSAT and Glucose 4 - may not accurately represent the performance of modern state-of-the-art SAT solvers. All solvers are evaluated on the same hardware but note that QuerySAT utilizes a single Nvidia T4 GPU while the others do not. We configure all three solvers to have approximately a 2-second time limit by giving QuerySAT 1024 recurrent steps, GSAT -- 500k steps, and setting a 2-second timeout for Glucose solver. We compare these solvers on previously described test sets of 3-SAT and SHA-1 tasks. The QuerySAT was trained on the forenamed train sets. The results are depicted as cactus plots in the Fig. \\ref{fig:sha1_query_vs_gsat} and \\ref{fig:scatter_query_vs_gsat}. Glucose utilizes the structure of the problem and therefore struggles on random 3-SAT instances yet solves the SHA-1 task in almost no time. Even though GSAT outperforms QuerySAT on 3-SAT instances, QuerySAT seems to utilize the SHA-1 structure and achieves better performance than GSAT. QuerySAT, similarly to other contemporary end-to-end neural solvers \\cite{selsam2018learning, amizadeh2019pdp}, in the general case, is outperformed by classical solvers and requires breakthroughs to allow scaling them to large industrial instances.\n\n\n\n\n\\section{Evaluating the query mechanism}\\label{sec:eval_query_mechanism}\n\n\n\\begin{table*}[ht]\n\\centering\n\\caption{Mean accuracy of NeuroCore and its augmentation with query and gradient and as per cent of fully solved instances from the test set over 3 consecutive runs. All models are trained with 32 recurrent steps and evaluated with 32, 512, and 4096 steps. Value after $\\pm$ indicates the standard error. Note that the training, validation and test instances are smaller than in Table \\ref{tab:sat_tasks_results}. \n}\n\\label{tab:query_generalization}\n\\begin{adjustbox}{max width=\\textwidth}\n\\begin{tabular}{lcccccc} \n\\toprule\n\\multirow{2}{*}{}                        & \\multicolumn{3}{c}{k-SAT  }                                                 & \\multicolumn{3}{c}{3-Clique}              \\\\ \n\\cmidrule(lr){2-4}\\cmidrule(lr){5-7}\n                                              & \\multicolumn{1}{c}{$s_{test} = 32$} & \\multicolumn{1}{c}{$s_{test} = 512$} & \\multicolumn{1}{c}{$s_{test} = 4096$} & \\multicolumn{1}{c}{$s_{test} = 32$} & \\multicolumn{1}{c}{$s_{test} = 512$} & \\multicolumn{1}{c}{$s_{test} = 4096$} \\\\ \n\\toprule\n\\textit{NeuroCore}\n&\\begin{tabular}{@{}c@{}}$38.86 \\pm 1.04$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$56.14 \\pm 3.48$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$60.97 \\pm 6.19$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$65.38 \\pm 5.09$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$67.12 \\pm 3.25$\\end{tabular} \n&\\begin{tabular}{@{}c@{}}$70.39 \\pm 2.68$\\end{tabular} \\\\\n\\midrule\n\\textit{\\, + Query}\n&\\begin{tabular}{@{}c@{}}$41.09 \\pm 2.33$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$61.21 \\pm 9.89$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$67.21 \\pm 13.54$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$62 .90 \\pm 2.55$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$78.19 \\pm 2.67$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$84.55 \\pm 2.93$\\end{tabular} \\\\\n\\midrule\n\\textit{\\, + Query + G}\n&\\begin{tabular}{@{}c@{}}$\\boldsymbol{47.92 \\pm 1.66}$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$\\boldsymbol{71.95 \\pm 6.59}$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$\\boldsymbol{75.50  \\pm 7.15}$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$\\boldsymbol{86.13 \\pm 4.83}$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$\\boldsymbol{93.96\\pm 2.59}$\\end{tabular}\n&\\begin{tabular}{@{}c@{}}$\\boldsymbol{95.50\\pm 1.95}$\\end{tabular} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{table*}\n\n\nWe evaluate the impact of the query mechanism by augmenting NeuroCore architecture with it and measuring the improvement. It is straightforward to do since NeuroCore is similar to QuerySAT. NeuroCore uses literals-to-clauses and clauses-to-literals message-passing to update the internal states for literals and clauses. Therefore, we add a query mechanism alongside the literals-to-clauses message-passing and also give the gradient of the evaluated query to the MLP that updates the literals state.  We experiment with two variants of the query mechanism: with both query and gradient (\\textit{+ Query + G}) and only with query (\\textit{+ Query}). Both versions are compared with the standard NeuroCore architecture (\\textit{NeuroCore}).\n\nWe chose k-SAT and 3-Clique tasks (see section \\ref{sec:querysat_evaluation}) for evaluation. We use the same dataset split as previously. For the k-SAT task, we use the same train and validation set as previously, but for the test set generate formulas with 100 to 200 variables. On the same note, train and validation sets for the 3-Clique task consists of graphs with 4 to 20 vertices but the test set of 20 to 40 vertices. The test set consists of harder formulas to see how various variants of query mechanism impacts generalization. \n\n\nAll three versions are trained for 100k iterations with the same training and network configuration as described in Section \\ref{sec:querysat_evaluation}. The trained model is then tested with 32, 512, and 4096 recurrent steps. The mean results of 3 consecutive runs are depicted in Table \\ref{tab:query_generalization}. We also include validation error in the train time for each version using 64 recurrent steps ($s_{test}$), it is depicted in the Fig. \\ref{fig:training_k-sat} and \\ref{fig:training_k-clique}.\n\nIn Table \\ref{tab:query_generalization}, we can see that both versions with a query mechanism outperform the NeuroCore baseline. The version with a query mechanism and gradient trains faster and achieves better accuracy. Interestingly, adding a gradient significantly improves the model's accuracy of the 3-Clique task. \nWe reason that gradient is helpful for the tasks that represented as a SAT instance has a distinct structure. Such structure is very expressive for the 3-Clique task, yet, on the contrary, k-SAT doesn't have any distinct structure as it is sampled from a uniform distribution \\cite{yolcu2019learning}. \n\n\n\n\\section{Related work}\nNeural networks have been proposed as an effective alternative for automatically developing heuristic algorithms for NP-hard problems \\cite{bengio2020machine, cappart2021combinatorial}. Two main research directions are replacing handcrafted heuristics with a neural network in a classical solver and building end-to-end neural solvers.\n\n\\cite{selsam2018learning} proposed a Graph Neural Network (GNN) architecture called NeuroSAT that is trained using single-bit supervision to predict whether the Boolean formula is satisfiable or not. The variable assignment is obtained from the last layer by performing  2-clustering on the output, but the network is never optimized for producing the assignment explicitly. Hence, it is not clear why such an approach should work.  In a later work, \\cite{selsam2019guiding} simplified NeuroSAT architecture (they call it NeuroCore)  and used it for guiding high-performance solvers (e.g.,  MiniSAT, Glucose) by predicting how likely each variable is in the unsatisfiable core.  \\cite{kurin2019improving} use Q-learning to train similar GNN for predicting branching heuristics in MiniSAT solver. Others \\cite{azar2020nngsat, kurin2019improving} similarly augment classical solvers with neural networks to solve 2-Quantified Boolean Formulas and logic locked circuits.  On the same trend, \\cite{yolcu2019learning} trained a GNN using reinforcement learning to learn a local-search heuristic that finds a solution similar to the GSAT \/ WalkSAT algorithm by flipping a single variable in each step. It is also theoretically shown that GNN can learn to mimic the WalkSAT algorithm \\cite{chen2019graph}. \n\nSupervised and reinforcement learning are not the best options for solving SAT, as the variable assignment is time-consuming to obtain, many possible solutions exist, and training may be slow. \\cite{amizadeh2018learning} proposed a differentiable unsupervised approach for directly solving Circuit-SAT. For this purpose, they represent variables as real numbers in the range $[0,1]$ and use GNN for producing variable predictions on the Circuit-SAT graph. The predicted variables are evaluated as Circuit-SAT, but the AND and OR functions are substituted with differentiable Softmax and Softmin functions. In a later work \\cite{amizadeh2019pdp}, they have applied the same method for solving SAT problems. To encourage exploration, they introduce a temperature parameter that is annealed in the training time, making training cumbersome. Although the training method is closely related to ours, their loss function is more complex.  \\cite{kyrillidis2020fouriersat} lately proposed to represent Boolean functions by multilinear polynomials and find a solution to a single formula by using gradient descent optimization. Even though their loss function is similar to ours, they solve the MAX-SAT problem and do not directly optimize towards finding a solution to the SAT problem.\n\n\n\\section{Conclusions}\nIn this paper, we have proposed a query mechanism that allows the neural network to make several solution trials, obtain the loss of each trial and change its strategy accordingly. To evaluate the impact of the query mechanism, we propose an unsupervised SAT loss and integrate it with queries to form the QuerySAT architecture. We find that QuerySAT outperforms the message-passing neural baseline on all proposed tasks: k-SAT, 3-SAT, 3-Clique, k-Coloring, and SHA-1 preimage attack. Experiments show that query mechanism can significantly increase the performance of message-passing graph neural networks. To give a better insight into the current capabilities of QuerySAT, we also compare it with classical solvers on 3-SAT and SHA-1 preimage attack tasks. Although we have analyzed only SAT solving, we expect a similar benefit from the query mechanism on other neural architectures and tasks. Since QuerySAT employs unsupervised loss, not requiring to know the labels, it provides rich opportunities for integration with classical solvers.\n\n\\section*{Acknowledgment}\n\n We would like to thank the IMCS UL Scientific Cloud for the computing power and Leo Truk\u0161\u0101ns for the technical support. This research is supported by Google Cloud and funded by the Latvian Council of Science, projects No.~lzp\\nobreakdash-2018\/1\\nobreakdash-0327, lzp-2021\/1-0479.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Motivations and outline}\nGibbs measures are familiar objects in various areas of applied\nprobability. Originally they have been devised in the framework of \nlattice spin systems to describe thermodynamic equilibrium states. \nObtaining these probability measures involves two basic steps. \nFirst a family of Gibbs measures for a finite number of random\nvariables is constructed as a modification of a reference measure \nusually describing the independent random field. The density is the\nexponential of an additive functional dependent on an interaction \nfunction. Then one takes a weak limit of these measures by increasing \nthe number of variables to infinity. This procedure copes with the\nfact that in usual model systems the interaction diverges in this\nlimit, thus the limit measure cannot be directly defined. \n\nThe context in which the class of Gibbs probability measures proved \nto be useful has substantially widened over the years, reaching the \nrealm of Brownian motion. \nOne natural way leading to Gibbs measures on path space is the \napplication of the Feynman-Kac formula and its extensions. Suppose \n$H$ is a self-adjoint operator acting on a Hilbert space $\\cal H$, and \nlet $f \\in {\\cal H}$. Then for a variety of specific choices of $H$ \nan equality of the type \n\\begin{equation}\n(e^{-tH}f)(x) = \\int_{\\mathbb{R}^d} dy \\int f(X_t) d\\mu_{[0,t]}(X|x,y)\n\\label{fk}\n\\end{equation}\ncan be derived, where $\\mathbb{R} \\ni t\\mapsto X_t \\in \\mathbb{R}^d$ is Brownian \nmotion and\n\\begin{equation}\nd\\mu_{[0,t]}(X|x,y) = e^{-U_t(X|x,y)} d\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_{[0,t]}(X).\n\\label{gibb}\n\\end{equation}\nHere $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_{[0,t]}$ is Brownian bridge over the bounded time \ninterval $[0,t]$, starting at $X_0 = x$ and ending at $X_t = y$, and \n$U_t$ is a functional of Brownian paths derived from $H$. After \nnormalizing to 1, the measure $\\mu_{[0,t]}(X|x,y)$ can be viewn as a \nGibbs measure on path space for ``finite volume'' $[0,t]$,  \n``interaction'' $U$, ``boundary condition'' $x,y$, and ``reference \nmeasure'' $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}$.  \n\nIn this paper we are interested in the question whether an extension \nin a suitable sense over the full time line $\\mathbb{R}$ of $\\mu_{[0,t]}$ \nexists. The details of the proof strongly depend on the choice of $U$, \nwhich for our purposes here will be specified below. These choices are \nmotivated by particular applications covered by the following classes\nof models, of which we talk only in this introduction. \n\n\\medskip\n\\vspace{0.1cm}\n\\noindent\nI. Densities dependent on the local time of Brownian motion\n\\begin{enumerate}\n\\item[(1)]\n{\\it $P(\\phi)_1$-process (It\\^o diffusion)} \\quad \nThis includes the familiar case of Schr\\\"odinger operators $H = \n(-1\/2)\\Delta + V$, ${\\cal H} = C_0^\\infty(\\mathbb{R}^d)$, with a potential \n$V(x)$ that can be chosen fairly generally (Kato-class). The \nwell-known result is \n$$\nU_t(X) = \\int_0^t V(X_s) ds.\n$$\n$\\mu$ describes in this case the path measure of the Markov process \ngiven by \n$$\ndX_t = dB_t + (\\nabla\\log\\Psi)(X_t)dt,\n$$\nwhere $B_t$ is $\\mathbb{R}^d$-valued Brownian motion and $\\Psi$ is the\neigenfunction of $H$ lying at the bottom of its spectrum (ground\nstate). It\\^o diffusions can be defined also on infinite dimensional \nspaces related to other SPDE. For further details on these Gibbs \nmeasures for bounded intervals and their extensions to $\\mathbb{R}$ we refer \nto \\cite{Sim75,Sim82,BL}.\n\n\n\\item[(2)] \n{\\it Nelson's model} \\quad \nThis is a scalar quantum field model describing the interaction \nof an electrically charged spinless particle with a boson field. In\nthis case $H$ is written as the sum of the free particle Hamiltonian\n$(-1\/2)\\Delta + V$, the free field Hamiltonian $\\int |k|a(k)^*a(k)dk$\nwith the usual boson creation and annihilation operators $a^*$ and $a$,\nand the interaction Hamiltonian $\\int(\\widehat\\rho(k)\/{\\sqrt{2|k|}})\n(e^{ik\\cdot x}a(k) + e^{-ik\\cdot x}a^*(k))dk$, with charge\ndistribution function $\\rho$. Moreover, ${\\cal H} = L^2(\\mathbb{R}^d,dx)\n\\otimes \\mathop{\\mathrm{{\\cal F}}}\\nolimits$, where $\\mathop{\\mathrm{{\\cal F}}}\\nolimits$ is Fock space, and a Feynman-Kac-type formula \nas (\\ref{fk}) above can be obtained by mapping $\\cal H$ into a space \nof continuous functions through a joint use of the so called ground \nstate transform and Wiener-It\\^o isomorphism yielding\n$$\nU_t(X) = \\int_0^t V(X_s) ds + \\int_0^t \\int_0^t W^\\rho(X_s-X_r,s-r) \ndsdr,\n$$\nwith \n$$\nW^\\rho(x,s) = -\\frac{1}{4}\\int_{\\mathbb{R}^d} \\frac{|\\widehat \\rho(k)|^2}{|k|} \ne^{-i k\\cdot x - |k||s|} dk.\n$$\nExtending this Gibbs measure from $[0,t]$ to $\\mathbb{R}$ \nis in this case of special interest since it allows a direct \nexpression of the ground state of $H$ in terms of its Radon-Nikodym \nderivative with respect to an underlying product measure, which makes \na rigorous derivation and proof of ground state properties possible. \nFor the case of translation invariant models (involving $V \\equiv 0$) \nthere are only few results available. For details see \n\\cite{LM,LMS1,LMS2,BHLMS}. \n\n\\item[(3)]\n{\\it Polaron and bipolaron models}\\quad\nThe polaron is a ``dressed'' electron (i.e., embedded into an energy\ncloud) interacting with a phonon field (i.e., quantum particles\ncarrying the vibrational energy of an ionic crystal). In this case we\nhave similar operators acting on the same Hilbert space as above \nexcept that the dispersion relation $|k|$ is replaced by 1 in the\nfree field Hamiltonian, and $\\widehat\\rho\/\\sqrt{2|k|}$ in the \ninteraction term by $1\/|k|$. This leads to the same $U_t$ as in the \ncase of Nelson's model with \n$$\nW^{\\rm\\tiny{pol}}(x,s) = -\\frac{1}{4|x|}\\,e^{-|s|}.\n$$\nThe bipolaron differs by the fact that it consists of two dressed\nelectrons coupled to the same phonon field, which are repelling each\nother by Coulomb interaction. In this case \n$$\nU_t(X) = \\alpha^2 \\int_0^t\\int_0^t {\\cal E}^W(X_s,Y_r,s-r) dsdr\n- g\\int_0^t \\frac{ds}{|X_s-Y_s|},\n$$\nwhere \n$$\n{\\cal E}^W(X_s,Y_r,u) = W(X_s-X_r,u) + 2W(X_s-Y_r,u) + W(Y_s-Y_r,u)\n$$\nwith $W = W^{\\rm\\tiny{pol}}$, $\\alpha < 0$ being the polaron-phonon\ncoupling paramater and $g>0$ the strength of the Coulomb repulsion \nbetween the two polarons. Moreover, in this case the reference measure\nis a product of two independent Wiener measures. For literature see\n\\cite{DV,S,LM06}.\n\n\\item[(4)]\n{\\it Intersection local time (weakly self-avoiding polymer)}\\quad\nFormally, the densities are given by\n$$ \nU_t(X) = \\int_0^t \\int_0^t \\delta(X_s-X_r) ds dr, \n$$ \nmeant to describe a polymer model with short-range ``soft-core'' \ninteraction encouraging to avoid self-intersections. For $d=2, 3$ \nsee \\cite{MR853758,MR667754,MR609228,MR573702,MR1240717}. In \n\\cite{Symanzik:1969lr} it was proved that in $d=2$ the model can be \nrigorously defined after an additive renormalization and the so \nobtained measure is absolutely continuous with respect to Wiener measure \nin 2 dimensions. In $d=3$ the singularity of the energy $U_t$ is more \nserious but an additive renormalization still suffices; Westwater proved \nexistence of the Gibbs measure which, however, in this case is not \nabsolutely continuous with respect to Wiener measure. Other works \ninclude \\cite{MR729794,MR1329112,MR889758,MR942042}.\n\n\\end{enumerate}\n\n\\vspace{0.1cm}\n\\noindent\nII. Stochastic currents\n\\begin{enumerate}\n\\item[(1)]\n{\\it Nelson's model in point charge limit and Pauli-Fierz model} \\quad\nThe point charge limit of Nelson's model corresponds to the case of \nreplacing $\\rho$ above with a delta-function, while the Pauli-Fierz \nmodel is obtained by replacing the scalar boson field with a quantized\nMaxwell (vector) field. The function in the density in both cases \nformally becomes \n\\begin{equation}\nU_t(X) = \\int_0^t V(X_s) ds + \\int_0^t \\int_0^t W(X_s-X_r,s-r) \ndX_s \\cdot dX_r,\n\\label{doubleito}\n\\end{equation}\nwith a $W=W^\\delta$ and $W=W^{\\rm PF}$ we do not write explicitly \ndown here. The difference from the case above is that instead of \ndouble Riemann we have to deal with double It\\^o integrals. A\nsubstantial difficulty to solve in this case is the proper definition \nof the double integrals in the first place, which will be done below. \nFor the specific model applications see \\cite{GL} in which we perform \nan ultraviolet renormalization of Nelson's Hamiltonian by using path\nmeasures whose densities are of the above form, and \\cite{H,HL2} for the\nPauli-Fierz model. The similar point charge limit in the latter model \nis an open problem.\n\n\\item[(2)]\n{\\it Turbulent fluids} \\quad\nIn fluid dynamics the understanding prevails that fully developed \nturbulence should be described by a suitable measure over\ndivergence-free velocity fields $u(x)$. One way of modelling it starts \nfrom the assumption that the vorticity field $\\nabla \\wedge u(x)$ is \nconcentrated along Brownian curves $X_t \\in \\mathbb{R}^3$. Under the \nEulerian incompressible flow, the kinetic energy $(1\/2) \\int u(x)^2 \ndx$ is conserved. The formal expression of the total energy is\n\\begin{equation}\\label{3.11}\nU_t(X)= \\int_0^t \\int_0^t \\frac{1}{|X_t - X_s|} \\; dX_t \\cdot dX_s.\n\\end{equation}\nIn order to have $e^{-U_t(X)}$ as a well-defined random variable at\nall, \\cite{Ffil} imposed the condition that the Coulomb potential is \nmollified so that the fluid has finite static energy. For details we\nrefer to~\\cite{statvort,FGub,Nua,evolution,Ascona}. \n\\end{enumerate}\n\n\n\n\\vspace{0.1cm}\n\\noindent\n{III. Processes with jumps} \\quad \n\\begin{enumerate}\n\\item[]\nApplications include Hamiltonians with spin. Since spin is a discrete \nvariable, this case goes beyond stochastic integration with respect to\nBrownian motion alone. The spin paths $\\sigma_t$ are described via a \nPoisson process $N_t$, and we obtain\n$$\nU_t(X) = \\int_0^t V(X_s) ds + \\int_0^t a(X_t) \\circ dX_t - \n\\int_0^t S(X_t,\\sigma_t) dt +\\int_0^{t^+} \\Phi(X_t,-\\sigma_t) dN_t.\n$$\nFor more details and explicit formulas of the terms above see\n\\cite{Jona,HL1}.\n\\end{enumerate}\n\n\\medskip\nIn this paper we address the purely mathematical problem of existence\nand characterization of Gibbs measures with densities of the type \n(\\ref{doubleito}). An accompanying paper \\cite{GL} takes this further \nto an application to quantum field theory. \n\nThe problem can be formulated in more generality by adding a term \nto $U_t$ in (\\ref{doubleito}) taking into account the dependence on \noutside paths of paths run within bounded time intervals. Choose\nwithout loss a bounded interval in the form $[-T,T] \\subset \\mathbb{R}$ and \nset up Wiener measure $\\mathop{\\mathrm{{\\cal W}}}\\nolimits_T$ on it. Consider the energy functions \ncorresponding to $[-T,T]$ as follows: \n\\begin{eqnarray*}\n&&\nV_T(X) := \\int_{-T}^T  V(X_s) ds \\qquad \\mbox{for external \npotential $V$}\\\\\n\\bigskip\n&&\nW_T(X) :=  \\int_{-T}^T \\int_{-T}^T W(X_t-X_s,t-s) dX_t \\cdot dX_s \n\\qquad \\mbox{internal energy} \\\\ \n\\medskip\\bigskip\n&&\nW_T(X|Y):= W^+_T(X|Y) + W^{-}_T(X|Y) \\qquad \n\\mbox{interaction energy}\n\\end{eqnarray*}\nwhere the interaction energies come from a pair interaction \npotential $W$, and $W^+_T(X|Y) := \\int_T^\\infty dY_t \\int_{-T}^T \ndX_s W(X_s-Y_t,t-s)$ is a term accounting for the interaction of \npaths inside $[-T,T]$ with paths in $[T,\\infty)$, and $W^-_T(X|Y) \n:= \\int_{-\\infty}^{-T} dY_t \\int_{-T}^T dX_s W(X_s-Y_t,t-s)$ for \nexternal paths running in $(-\\infty,-T]$. As in (\\ref{gibb}), \nthese energies give rise to the Gibbs measure for $[-T,T]$\n\\begin{equation}\n\\label{ea:gibbs1}\nd\\mu_T(X) := \\frac{e^{-V_T(X)-\\lambda W_T(X)}}{Z_T}\\, d\\mathop{\\mathrm{{\\cal W}}}\\nolimits_T(X)\n\\end{equation}\nwith a parameter $\\lambda \\in \\mathbb{R}$ tuning the strength of the pair \npotential, and $Z_T$ the normalizing factor turning it into a\nprobability measure. The object of interest are the accumulation \npoints $\\mu$ of the family of measures $\\{\\mu_{T}\\}_{T>0}$ in the \ntopology of local weak convergence. In the DLR \n(Dobrushin-Lanford-Ruelle) approach, these limit points can be \ncharacterized by a property of consistency with respect to a \nprescribed family of probability kernels (specification) providing\nthe local conditional probabilities of the limit random fields. These \nkernels are given by \n\\begin{equation}\n\\label{eq:specc}\n\\mu_T(dX|Y) := \\frac{e^{-V_T(X)-\\lambda W_T(X)-\n\\lambda W_T(X|Y)}}{Z_T(Y)} \\, d\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{Y_{-T},Y_{T}}_T(X)\n\\end{equation}\nwith external (or boundary) path $Y$. The possible accumulation \npoints of the family $\\{\\mu_T\\}_T$ should then satisfy the DLR \nequations\n$\n\\int \\mu_T(A|Y) \\mu(dY) = \\mu(A)\n$ \nfor all cylinder sets $A$ in the sub-$\\sigma$-field generated by \nprojections to $[-T,T]$ for all $T>0$.\n\nIn contrast to the case of interactions depending on the local time \nof the process $X$ (given by double Riemann integrals listed above) \nour case encounters two difficulties.\n\\begin{itemize}\n\\item[(1)]  \nThe expressions of $W_T(X)$ and $W_T(X|Y)$ are only formal: \nThe double stochastic integrals are not well defined since the \nintegrands are neither forward nor backward adapted with respect \nto the semimartingale $X$ (Brownian bridge under $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_T$).\n\\item[(2)]  \nThe specification (\\ref{eq:specc}) must be defined pathwise for each \n$Y$, however, in general the only information we have on the boundary \npath $Y$ is that it is a continuous path with a Brownian-like\nregularity. This is insufficient for defining the line integrals with \nrespect to $dY$ appearing in $W_T(X|Y)$.\n\\end{itemize}\n\n\\noindent\nOf these two difficulties the first requires a minor amendment, \nwhile the second is far more serious and will urge us to introduce \nthe novel setting of \\emph{Brownian currents} in which we can make\nsense of a formulation of the DLR equations. Roughly speaking, we \nprovide each sample path with sufficient information in order to \ndetermine the work made by test vector fields. Specifically, we \nconsider the family of random variables $\\{X_t\\}_t$ jointly with \nthe random variables \n$\n C_{st}^X( \\varphi) := \\int_s^t \\varphi(u,X_u) dX_u  \n$\n(\\emph{stochastic currents})\nand show that in this augmented sample space the specification can \npathwise be defined. (This problem has some analogy with difficulties \nat defining specifications for unbounded spin systems for which too \none has to select a subset of admissible boundary conditions so that \nthe interaction energy makes sense. Our approach here goes further in \nthat we do not only have to control the growth of the boundary paths \nbut must also provide a priori the value of the line integrals against \na sufficiently large set of test vector fields $\\varphi$.) To perform \nthis ``lifting'' procedure we use techniques of rough paths theory. \n\n\\medskip\nHere is an outline of the paper. In Chapter 2 \nwe present our results on rough paths that will be put at use \nsubsequently. In Chapter 3 we introduce the framework of stochastic \ncurrents, in particular Brownian currents. In Chapter 4 we define\nGibbs measures on these currents for bounded intervals of the line. In \nChapter 5 we turn to proving that these Gibbs measures can be extended \nover the whole line provided the interaction term is weakly coupled. \nThis requirement is needed since, \\emph{faute de mieux}, we use cluster \nexpansion in order to construct weak limits of Gibbs measures for \nbounded intervals. The version of cluster expansion we develop here is \ndifferent from the conventional ones since the configurations are \nsegments of Brownian paths rather than spins of compact or real-valued \nstate space, the interactions depend on double stochastic integrals, \nthe reference measure is not a product measure, and the measure we want \nto construct is non-Markovian. However, it has the same spirit of usual \ncluster expansions as it splits off into a part of hands-on analysis of \nenergy bounds and a part of combinatorics. Beside a proof of \nexistence, cluster expansion allows us to study also uniqueness, typical\npath behaviour and mixing properties of Gibbs measures, which will be\ndone in Chapter 6. \n\n\n\n\\label{sec:gibbs}\n\n\\section{Rough paths}\n\\subsection{Definition of double stochastic integrals by rough paths}\n\\label{sec:rough-paths}\n\nIt\\^o's theory of integration gives a meaning to line integrals of the \nform \n$\n\\int_{-T}^T \\varphi(t,X_t) dX_t,\n$\nwith $X$ a semimartingale (with respect to its natural forward\nfiltration $\\mathcal{F}$) and $\\varphi(t,\\xi)$ some function. The way\nof obtaining it is by taking limits of Riemann sums\n$\n\\sum_\\alpha \\varphi(\\tau_\\alpha,X_{\\tau_\\alpha}) \n(X_{\\tau_{\\alpha+1}}-X_{\\tau_\\alpha})\n$\nover a family of partitions $\\{\\tau_\\alpha\\}_\\alpha$ of $[-T,T]$ with \nmesh decreasing to zero. Such limits are known to exist whenever \n$\\varphi(t,X_t)$ is an $\\mathcal{F}$-adapted functional of $X$, i.e.,\nif it only depends on $\\mathop{\\mathrm{{\\cal F}}}\\nolimits_t$ for any $t\\in[-T,T]$ and has some \nintegrability properties. Then \n$\n\\int_{-T}^T \\varphi(t,X_t) dX_t\n$\nis defined as a random variable on the same probability space on which \n$X$ is defined. In our applications $X$ is the coordinate process on the\nBorel space $\\mathcal{X}$, and the stochastic integral defines a Borel map from\n$\\mathcal{X}$ to $\\mathbb{R}$ which is defined on a set of full $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$ measure.\nIt is important to remember the crucial fact that the full-measure set \nin general depends on the integrand: different integrands may give \ndifferent full-measure sets.\n\nIn this section we analyze the regularity of such integrals by using\nthe theory or rough paths as developed in \n\\cite{lyons-98,lyons-qian-02,MR2091358,feyel}. This theory allows to \nproperly understand stochastic integrals from a purely analytic \nperspective. To gain this freedom we need to enlarge the sample space \nover which the measures are defined, however, this will turn out to \nmake no harm in our applications.\n\nRough paths theory has been devised by T.~Lyons~\\cite{lyons-98} to\ngive a meaning to line integrals of the form\n\\begin{equation}\n  \\label{eq:basic-rp-integral}\n\\int_0^T \\varphi(X_t) dX_t,  \n\\end{equation}\nin case of $X$ being an irregular function of the parameter. A typical\ncase of interest is when $X$ is chosen to be H\\\"older continuous with \nsmall exponent, such as $\\gamma\\in(0,1)$. \nThe natural approach to defining such integrals is that of taking\nRiemann approximations over a finite partition of $[0,T]$ and proving \nthat the sequence converges as the mesh of the partition goes to zero. \nWhen general integrals in the form $\\int_0^T Y_s dX_s$ are considered\nwith bounded $Y$, this can only hold if $X$ is a process of bounded \nvariation leading to the familiar Stieltjes integral. When $Y$ is \nH\\\"older continuous with exponent $\\rho$, Young's work \\cite{young-36} \nmakes sure that a sufficient condition for the convergence of the \nRiemann sums is that $\\gamma+\\rho > 1$. The Young integral is useful \nin studying, for example, fractional Brownian motion of Hurst index \n$H>1\/2$, which is a stochastic process with H\\\"older continuous paths \nof any exponent $\\gamma < H$. In this case (\\ref{eq:basic-rp-integral}) \ncan be defined for any function $\\varphi$ which is at least $C^1$.\n\nIntegrals of the form (\\ref{eq:basic-rp-integral}) when $X$ is a\nsample of Brownian motion are not within the reach of Young's theory \nand indeed it is not difficult to see that different definitions of \nthe Riemann approximations can lead to different limits (or even not \nconverge at all). This difficulty lies at the basis of the existence \nof more than one type of stochastic integral over Brownian motion, \ntwo well known possibilities being the integrals developed by It\\^o \nand Stratonovich. Lyons's approach to the problem was proving that \nunder reasonable conditions the Riemann sums for $\\int_0^T \\varphi(X_t) \ndX_t$ can be modified by adding an extra term in order to make them\nconverge. This compensation is not unique but in many relevant cases \nit can be performed in such a way that the so obtained integral is an \nextension of the classical line integral and Young integral. That is,\nwhenever $X$ is regular enough, the modified integral coincides with \nthe Young one or Riemann-Lebesgue integral if moreover $X$ is almost \nsurely is differentiable.\n\nWe consider only the case when $X$ is H\\\"older continuous with $\\gamma \n> 1\/3$ since this includes the case of Brownian motion and all our \npresent applications. (The theory for more general $\\gamma$ would be \nfar more cumbersome.) We use the notation $X_{st} = X_t-X_s$, and make \nthe following basic\n\n\\begin{definition}\nFor each bounded $I \\subset \\mathbb{R}$ we call the couple $(X,\\mathbb{X})$  \na \\emph{2-step rough path}, where\n\\begin{enumerate}\n\\item[(1)]\n$X \\in C^\\gamma(I,\\mathbb{R}^d)$;\n\n\\item[(2)\nthere exists a function $\\mathbb{X} : I\\times I \\to \\mathbb{R}^{d}\\times \\mathbb{R}^d$ \nsatisfying the \\emph{multiplicative property}\n  \\begin{equation}\n    \\label{eq:mult}\n  \\mathbb{X}^{ij}_{st} - \\mathbb{X}^{ij}_{su} - \\mathbb{X}^{ij}_{ut} = \n  X^i_{su} X^j_{ut}   \n  \\end{equation}\nfor all $s \\le u \\le t \\in I$ and any $i,j =1,\\dots,d$;\n\n\\item[(3)]\nthere is $C \\in [0,\\infty)$ such that\n\\begin{equation}\n  \\label{eq:rough-hold-area}\n\\|\\mathbb{X}_{st}\\|_{2\\gamma} := \\sup_{t \\neq s} \n\\frac{|\\mathbb{X}_{st}|}{|t-s|^{2\\gamma}} \\le C.\n\\end{equation}\n\\end{enumerate}\n\\end{definition}\n\\noindent\nWe use the similar norm $\\|{X_{st}}\\|_\\gamma$ correspondingly given\nby the right hand side of (\\ref{eq:rough-hold-area}).\n\n\\begin{remark}\n\\rm{\nNote that Assumption\n(2) is non-trivial only if $d>1$. When $d=1$ we can take\n$\n\\mathbb{X}_{st} = (X_{st})^2\/2,\n$\nand it can easily be seen that both (\\ref{eq:mult}) and the\nH\\\"older-like bound~(\\ref{eq:rough-hold-area}) are then satisfied. \n}\n\\end{remark}\n\n\nFor our purposes below we need a slight extension since we want \nto be able to integrate functions explicitly dependent on the time\nparameter, such as $\\int_0^T \\varphi(u,X_u) dX_u$. One possibility \nis to consider the couple $(u,X)$ as a new path and construct the \nassociated rough path. This construction, however, has the disadvantage \nof requiring rather much regularity of the function $\\varphi$. A more \neffective approach is to treat the line-integral as a Young integral \nwith respect to the $u$ dependence of the integrand as soon as the \nmap $u\\mapsto \\varphi(u,\\xi)$ has H\\\"older exponent greater than \n$1-\\gamma$ (as we shall see below).\n\nFor the (time-dependent) test vector fields in $C(\\mathbb{R} \\times \\mathbb{R}^d; \\mathbb{R}^d)$\n we use the norm\n$$\n\\|\\varphi\\|_{\\rho,2,s,t} = \\sup_x \\left[ \\sup_{u\\in[s,t]}\n\\left(\\max_{k=0,1,2} |\\nabla^k \\varphi(u,x)|\\right) +\n \\sup_{u,v \\in[s,t]} \\max_{k=0,1}\n\\frac{|\\nabla^k \\varphi(u,x)-\\nabla^k \\varphi(v,x)|}{|u-v|^\\rho} \n\\right]\n$$\nwith $\\rho > 0$ and the convention $\\nabla^0 \\varphi(t,x) = \n\\varphi(t,x)$. \nThen our basic result on the step-2 rough path $(X,\\mathbb{X})$  is\n\n\\begin{theorem}\n\\label{th:rough-main}\nLet $\\varphi \\in C(\\mathbb{R} \\times \\mathbb{R}^d,\\mathbb{R}^d)$ be such that it is $C^2$ \nwith respect to its second variable and H\\\"older continuous with\nexponent $\\rho$ with respect to its first variable, such that \n$\\rho+\\gamma > 1$. Then the sums\n$$\n\\sum_\\alpha \\left(\\varphi_i(\\tau_\\alpha,X_{\\tau_\\alpha}) \nX^i_{\\tau_{\\alpha+1} \\tau_\\alpha}+\\nabla_j \n\\varphi_i(\\tau_\\alpha,X_{\\tau_\\alpha}) \n\\mathbb{X}_{ \\tau_\\alpha \\tau_{\\alpha+1}}^{ij} \\right), \n\\quad i,j=1,...,d\n$$\nconverge as the mesh of the partition $\\{\\tau_\\alpha\\}_{\\alpha}$ of\n$[0,T]$ goes to zero, and defines the integral\n$\n\\int_0^T \\varphi(u,X_u) dX_u \n$ \nMoreover for any $T\\le 1$ we have the bound\n$$\n\\left|\\int_0^T \\varphi(u,X_u) dX_u \\right| \\le C T^\\gamma  \n\\|\\varphi\\|_{\\rho,2,0,T} (1+\\|X\\|_\\gamma+\\|\\mathbb{X}^2\\|_{2\\gamma})^3.\n$$\n\\end{theorem}\n\\begin{proof}\nWe prove convergence over the dyadic partition of $[0,1]$, then \nconvergence of a general partition follows then by the arguments \ndeveloped in~\\cite{MR2091358}. Let $\\tau^{(n)}_{\\alpha}=T \\alpha\/2^n$ \nfor $\\alpha = 0,\\dots,2^n$, and let\n$$\nS_n = \\sum_{\\alpha=0}^{2^n-1} \n\\left(\\varphi_i(\\tau^{(n)}_\\alpha,X_{\\tau^{(n)}_\\alpha}) \nX^i_{\\tau^{(n)}_{\\alpha+1}\n\\tau^{(n)}_\\alpha}+\\nabla_j \n\\varphi_i(\\tau^{(n)}_\\alpha,X_{\\tau^{(n)}_\\alpha}) \n\\mathbb{X}_{\\tau^{(n)}_{\\alpha+1}\n  \\tau^{(n)}_\\alpha}^{ij}\\right). \n$$\nNote that\n\\begin{equation}\n\\label{eq:rough-split}\n S_{n}-S_{n-1} = \\sum_{\\alpha=0}^{2^n-1} \n(A_\\alpha^1 + A_\\alpha^2  +\n A_\\alpha^3 + A_\\alpha^4),\n\\end{equation}\nwhere\n\\begin{eqnarray*}\n&& \nA^1_\\alpha  =\n\\left[\\varphi_i(\\tau^{(n)}_{2\\alpha+1},X_{\\tau^{(n)}_{2\\alpha+2}})-\n\\varphi_i(\\tau^{(n)}_{2\\alpha+1},X_{\\tau^{(n)}_{2\\alpha+1}})-\\nabla_j\n\\varphi_i(\\tau^{(n)}_{2\\alpha+1},X_{\\tau^{(n)}_{2\\alpha+1}}) \nX^j_{\\tau^{(n)}_{2\\alpha+2} \\tau^{(n)}_{2\\alpha+1}} \\right] \nX^i_{\\tau^{(n)}_{2\\alpha+1} \\tau^{(n)}_{2\\alpha}}, \\\\\n&& \nA^2_\\alpha  =\n\\left[\\nabla_j\\varphi_i(\\tau^{(n)}_{2\\alpha+1},X_{\\tau^{(n)}_{2\\alpha+2}})- \n\\nabla_j \\varphi_i(\\tau^{(n)}_{2\\alpha+1},X_{\\tau^{(n)}_{2\\alpha+1}})\\right] \n\\mathbb{X}_{\\tau^{(n)}_{2\\alpha+1} \\tau^{(n)}_{2\\alpha}}^{ij},\\\\\n&&\nA^3_\\alpha  = \n\\left[\\varphi_i(\\tau^{(n)}_{2\\alpha+2},X_{\\tau^{(n)}_{2\\alpha+2}})-\n\\varphi_i(\\tau^{(n)}_{2\\alpha+1},X_{\\tau^{(n)}_{2\\alpha+2}}) \\right] \nX^i_{\\tau^{(n)}_{2\\alpha+1}\\tau^{(n)}_{2\\alpha}},\\\\\n&& \nA^4_\\alpha  = \n(\\nabla_j\\varphi_i(\\tau^{(n)}_{2\\alpha+2},X_{\\tau^{(n)}_{2\\alpha+2}})-\n\\nabla_j \\varphi_i(\\tau^{(n)}_{2\\alpha+1},X_{\\tau^{(n)}_{2\\alpha+2}})) \n\\mathbb{X}_{\\tau^{(n)}_{2\\alpha+1}\\tau^{(n)}_{2\\alpha}}^{ij}.\n\\end{eqnarray*}\nBy using the fact that $\\varphi\\in C^2$, we have the bounds \n$\n|A^1_\\alpha | \\le T^{3\\gamma}\\|\\varphi\\|_{\\rho,2,0,T} \\|X\\|^3_\\gamma\n2^{-3\\gamma n}\n$\nand\n$\n|A^2_\\alpha| \\le  T^{3\\gamma} \\|\\varphi\\|_{\\rho,2,0,T} \\|X\\|_\\gamma\n\\|\\mathbb{X}\\|_{2\\gamma} 2^{-3\\gamma n}\n$\non the first two terms.\nFor the last two we have\n$\n|A^3_\\alpha| \\le  T^{\\rho+\\gamma} \\|\\varphi\\|_{\\rho,2,0,T}\n\\|X\\|_\\gamma 2^{-n(\\rho+\\gamma)}\n$\nand \n$\n| A^4_\\alpha | \\le  T^{\\rho+2\\gamma} \\|\\varphi\\|_{\\rho,2,0,T}\n\\|\\mathbb{X}\\|_{2\\gamma} 2^{-n(\\rho+2\\gamma)}\n$.\nThus\n$$\n|S_{n}-S_{n-1}| \\; \\le \\; C (T^{3\\gamma} +\nT^{\\rho+\\gamma})\\|\\varphi\\|_{\\rho,2,0,T} (2^{(1-3\\gamma) n}\n+ 2^{(1-\\gamma-\\rho)n}) \n$$\nwhere $C$ is a function dependent only on $X,\\mathbb{X}$. Write\n$S_n$ as the telescopic sum \n$\nS_n = S_0 + \\sum_{k=1}^n(S_k-S_{k-1})\n$.\nThen, since\n$$\n\\sum_{k=1}^n |S_k-S_{k-1}| \\le C (T^{3\\gamma}+T^{\\rho+\\gamma}) \n\\|\\varphi\\|_{\\rho,2,0,T} \\sum_{k=1}^n (2^{(1-3\\gamma) n}+2^{(1-\\gamma-\\rho) n})\n$$\nconverges geometrically whenever $\\gamma > 1\/3$ and $\\gamma+\\rho>1$,\nwe obtain absolute convergence of the sequence $S_n$. The bound on the \nintegral thus follows.\n\\end{proof}\n\n\\medskip\nAs this theorem shows, by using the additional information provided by \nthe path $\\mathbb{X}$ we are able to make Riemann sums converge. The \nfollowing consequence is immediate.\n\\begin{corollary}\nThe integral defined in Theorem \\ref{th:rough-main} has\n$\n\\mathbb{X}_{st}^{ij} = \\int_{s}^t X^i_{su} dX^j_u\n$.  \n\\end{corollary}\n\\begin{proof}\nWe have\n$$\n\\int_{s}^t X^{i}_{us} dX^j_u = \\int_{s}^t X^i_{u} dX^j_u - X^i_s \n(X^j_u-X^j_s),\n$$ \nas it is easily seen by using the definition. That is, the rough \nintegral has the same linearity property as Riemann integral and\nbehaves the same way with respect to the integration of constants. \nMoreover, the integral in the right hand side corresponds to the \nfunction $\\varphi_k(\\xi) = \\xi^i \\delta_{kj}$ and by using \n(\\ref{eq:rough-split}) we have $S_{n}=S_{n-1}$ on the dyadic partition \nabove. Hence $S_n = S_0$ and \n$$\n\\int_{s}^t X^i_{u} dX^j_u = S_0 = \\sum_k \\varphi_k(X_s) X^k_{st} +\n\\sum_{k,m} \\nabla_m \\varphi_k(X_s) \\mathbb{X}_{st}^{km} = X_s^i\nX^j_{st} + \\mathbb{X}_{st}^{ij},\n$$\nwhich completes the proof.\n\\end{proof}\n\n\\begin{remark}\n\\rm{\nThe function $\\mathbb{X}$ can be identified as giving the value of a \n\\emph{twice iterated integral} over $X$,\n\\begin{equation}\n  \\label{eq:rough-ident}\n\\mathbb{X}_{st}^{ij} = \\int_s^t \\left(\\int_s^u dX^i_v\\right) dX^j_u. \n\\end{equation}\nThus Theorem \\ref{th:rough-main} can be alternatively interpreted as \nsaying that the knowledge of the twice iterated integral (in addition \nwith some H\\\"older continuity) is sufficient for determining the value \nof the integral $\\int \\varphi(X_t)dX_t$ for arbitrary $C^2$ function.  \n}\n\\end{remark}\n\nProvided $\\gamma > 1\/2$ and (\\ref{eq:rough-ident}) holds, whenever the\nright hand side is defined by using the Young integral, the integral \ndefined in Theorem \\ref{th:rough-main} coincides with the Young\nintegral. This is easy to see due to the estimate \n$$\n|\\mathbb{X}_{st}^{ij}| \\le C (\\|X\\|_{\\infty}+\\|X\\|_\\gamma)^2 \n|t-s|^{2\\gamma}\n$$\nfor the Young integral. Since $2\\gamma > 1 $, the sums\n$\n\\sum_\\alpha \\nabla_j \\varphi_i(X_{\\tau_\\alpha}) \n\\mathbb{X}_{\\tau_{\\alpha+1} \\tau_\\alpha}^{ij} \n$\nvanish as the mesh goes to zero, so the modified and Riemann sums \nconverge to the same limit.\nMoreover, the rough integral is continuous in the natural topology\nassociated to the step-2 rough path $(X,\\mathbb{X})$, i.e., we have\n\\begin{corollary}\nLet $(X^n,\\mathbb{X}^n)$ be a sequence of step-2 rough paths such \nthat\n$\n\\|X^n - X\\|_\\gamma + \\|\\mathbb{X}^n-\\mathbb{X}\\|_{2\\gamma} \\to 0.\n$  \nThen\n$\n\\int \\varphi(X^n) dX^n \\to \\int \\varphi(X) dX,\n$\nfor all $\\varphi \\in C^2$.\n\\end{corollary}\n\nSuppose $X^n$ is a sequence of regular (say, piecewise linear)\napproximations of the H\\\"older continuous path $X$. Then by putting \n$\\mathbb{X}^n = \\int \\int dX^n\\otimes  dX^n $, where the integrals \nare Riemann integrals, a sufficient condition for the convergence of \nthe approximate integrals $\\int \\varphi(X^n) dX^n$ to $\\int \\varphi(X) \ndX$ is that the sequence $\\mathbb{X}^n$ converges to $\\mathbb{X}$ in \nthe topology induced by $\\|\\cdot\\|_{2\\gamma}$.\n\n\n\\subsection{Brownian motion as rough path}\n\nLet now $X$ be a sample path of Brownian motion. By Kolmogorov's Lemma \nthe process $t \\mapsto X_t$ has a version that is H\\\"older continuous \nwith exponent $\\gamma \\in(1\/3,1\/2)$ (actually $\\gamma$ can be taken \narbitrarily close to $1\/2$). In the following we will use such a \nversion without each time mentioning explicitly, i.e., consider a \nsubspace $\\mathcal{X}_\\gamma \\subset C(\\mathbb{R},\\mathbb{R}^d)$ such that every $X \\in \n\\mathcal{X}_\\gamma$ is H\\\"older continuous with exponent $\\gamma$.\n \nTo apply the above results to $X$ we need a choice for $\\mathbb{X}$. \nThis candidate is not unique, different choices will lead to different \nintegrals over $X$. For instance, in order to construct a possible \n$\\mathbb{X}$ we can start by setting\n$$\n(\\mathbb{X}_{\\text{It\\^o}})^{ij}_{st}=\\int_s^t \\int_s^u dX^i_v dX^j_u,\n$$  \nwhere the double integral is understood in It\\^o sense. In this way we\nobtain a family of random variables\n$\\{(\\mathbb{X}_{\\text{It\\^o}})^{ij}_{st}: i,j=1,\\dots,d; \\; t,s\n\\in[0,T]\\}$ satisfying the multiplicative property\n\\begin{equation*}\n(\\mathbb{X}_{\\text{It\\^o}})^{ij}_{st} - \n(\\mathbb{X}_{\\text{It\\^o}})^{ij}_{ut} -\n(\\mathbb{X}_{\\text{It\\^o}})^{ij}_{su} \n= X^i_{ut} X^j_{su}   \n\\end{equation*}\nalmost surely for any fixed $t,s,u\\in[0,T]$. The next step is to show \nthat this family has a version for which\n\\begin{equation}\n  \\label{eq:rough-holder-area}\n\\|\\mathbb{X}^{ij}_{\\text{It\\^o}}\\|_{2\\gamma} < \\infty  \\quad \\mbox{a.s.}\n\\end{equation}\nIn order to prove (\\ref{eq:rough-holder-area}), we use the following\nlemma obtained as an extension of a result of Garsia-Rodemich-Rumsey \nin \\cite{MR2091358}.\n\\begin{lemma}\n\\label{lemma:besov}\nFor any $\\theta > 0$ and $p \\ge 1$ there exists a constant $C$ such\nthat for any $R \\in C([0,T]^2,{\\cal B})$, where $({\\cal B},|\\cdot|)$ \nis a Banach space, we have \n\\begin{equation}\n\\label{eq:generalboundxx}\n\\|R\\|_{\\theta} \\le C \\left(U_{\\theta+2\/p,p}(R)+V_\\theta(R)\\right),\n\\end{equation}\nwith\n\\begin{equation*}\n U_{\\theta,p}(R) = \\left[ \\int_{[0,T]^2}\n \\left(\\frac{|R_{t s}|}{|t-s|^\\theta}\\right)^p dt ds \\right]^{1\/p}\n\\end{equation*}\nand\n\\begin{equation*}\n V_\\theta(R) = \\inf_{\\theta_1 \\in(0,\\theta)} \\sup_{t \\neq u \\neq s} \n\\frac{|R_{st}-R_{ut}-R_{su}|}{|t-u|^{\\theta_1}|u-s|^{\\theta-\\theta_1}}. \n\\end{equation*}\n\\end{lemma}\n\n\\begin{corollary}\nWe have $\\|\\mathbb{X}_{{\\text{It\\^o}}}\\|_{2\\gamma} < \\infty$ $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$-almost \nsurely.  \n\\end{corollary}\n\\begin{proof}\nConsider $V_{2\\gamma}(\\mathbb{X}_{\\text{It\\^o}})$. By using the \nmultiplicative property we have\n\\begin{eqnarray*}\nV_{2\\gamma}(\\mathbb{X}_{\\text{It\\^o}}) \n&\\le&  \n\\inf_{\\theta_1 \\in (0,2\\gamma)} \\sup_{t \\neq u \\neq s} \n\\frac{|X_{tu}||X_{us}|}{|t-u|^{\\theta_1}|u-s|^{2\\gamma-\\theta_1}}\\\\ \n&\\le&  \n\\inf_{\\theta_1 \\in(0,2\\gamma)} \\sup_{t \\neq u} \\frac{|X_{tu}|}\n{|t-u|^{\\theta_1}}\\sup_{u \\neq s}\n\\frac{|X_{us}|}{|u-s|^{2\\gamma-\\theta_1}}\\\\\n&\\le&  \n\\left(\\sup_{t \\neq u} \\frac{|X_{tu}|}{|t-u|^{\\gamma}}\\right)^2\\\\\n& = &\n\\|X\\|_\\gamma^2.\n\\end{eqnarray*}\nMoreover,\n\\begin{eqnarray*}\n\\mathbb{E} [U_{2\\gamma,p}(\\mathbb{X}_{\\text{It\\^o}})^p] \n&=& \n\\mathbb{E}  \\left[\\int_{[0,T]^2}\\left(\\frac{|(\\mathbb{X}_{\\text{It\\^o}})_{t s}|}\n{|t-s|^{2\\gamma}}\\right)^p dt ds \\right]\\\\\n&=&    \n\\int_{[0,T]^2}\\frac{ \\mathbb{E} [|(\\mathbb{X}_{\\text{It\\^o}})_{t s}|^p]}\n{|t-s|^{2\\gamma p}} dt ds, \n\\end{eqnarray*}\nwith expectation with respect to Wiener measure. An application of the \nBurkholder-Davis-Gundy inequality~\\cite{Yor} allows to estimate the \n$p$-moment of the double stochastic integral as\n\\begin{eqnarray*}\n\\mathbb{E} [|(\\mathbb{X}_{\\text{It\\^o}})_{t s}|^p] \n& \\le&\nc_p \\mathbb{E} \\left[\\int_s^t |X_{us}|^2 ds\\right]^{p\/2} \\\\\n&\\le&\nc_p |t-s|^{p\/2-1} \\mathbb{E} \\left[\\int_s^t |X_{us}|^p ds\\right] \\\\ \n& \\le &\nc_p' |t-s|^{p\/2-1}\\int_s^t |u-s|^{p\/2} ds \\\\\n&\\le &\nc_p'' |t-s|^{p},   \n\\end{eqnarray*}\nfor all $p > 1$ and some $c_p, c_p', c_p'' > 0$. Thus\n\\begin{eqnarray*}\n\\mathbb{E} [U_{2\\gamma,p}(\\mathbb{X}_{\\text{It\\^o}})^p] \n&\\le& \nc_p'' \\int_{[0,T]^2} \\frac{ 1}{|t-s|^{(2\\gamma-1) p}} dt ds < \\infty,\n\\end{eqnarray*}\nfor any $\\gamma < 1\/2$, by choosing $p$ large enough ($p > 1\/(1-2\\gamma)$).\n\\end{proof}\n \nThis last result shows that there exists a version of the stochastic process\n$(X,\\mathbb{X}_{\\text{It\\^o}})$ that is a step-2 rough path; from now on we \ndenote by $(X,\\mathbb{X}_{\\text{It\\^o}})$ this particular version. Then \nintegrals can be defined by applying Theorem \\ref{th:rough-main}. We\ncall such an integral \\emph{rough integral} over $(X,\\mathbb{X}_{\\text{It\\^o}})$.\n\nThe relationship between the rough integral and the It\\^o integral is made\nclear by \n\n\\begin{lemma}\nThe rough integral over the couple $(X,\\mathbb{X}_{\\text{It\\^o}})$\ncoincides with the It\\^o integral for any $\\varphi \\in C^2$.  \n\\end{lemma}\n\\begin{proof}\nBy It\\^o theory the sums\n$\n\\sum_\\alpha \\varphi_i(X_{\\tau_\\alpha}) X^i_{\\tau_{\\alpha+1} \\tau_\\alpha}\n$ \nconverge in probability to the It\\^o integral $\\int_0^T \\varphi(X_u) dX_u$.\nHence it suffices to show that the sums\n$\n\\sum_\\alpha \\nabla_j \\varphi_i(X_{\\tau_\\alpha}) \\mathbb{X}_{\\tau_{\\alpha+1}\n  \\tau_\\alpha}^{ij} \n$\nconverge to zero in $L^2$ sense as then it follows that the two\nintegrals almost surely coincide. A simple computation shows that\n\\begin{equation}\n\\mathbb{E}\\left[ \\sum_\\alpha \\nabla_j \\varphi_i(X_{\\tau_\\alpha}) \n\\mathbb{X}_{\\tau_{\\alpha+1} \\tau_\\alpha}^{ij} \\right]^2 \n=  \n\\sum_\\alpha \\mathbb{E} \\left[  \\nabla_j \\varphi_i(X_{\\tau_\\alpha}) \n\\mathbb{X}_{\\tau_{\\alpha+1} \\tau_\\alpha}^{ij} \\right]^2,\n\\nonumber\n\\end{equation}\nsince the cross terms are all zero in the mean by independence of the\nincrements of the Brownian motion. Hence \n\\begin{equation*}\n  \\begin{split}\n& \\mathbb{E}\\left[ \\sum_\\alpha \\nabla_j \\varphi_i(X_{\\tau_\\alpha}) \n  \\mathbb{X}_{\\tau_{\\alpha+1} \\tau_\\alpha}^{ij} \\right]^2\n    \\le \\|\\varphi\\|_1 \\sum_\\alpha \\mathbb{E} \\left[ \\mathbb{X}_{\\tau_{\\alpha+1}\n  \\tau_\\alpha}^{ij} \\right]^2\n    \\le C \\|\\varphi\\|_1 \\sum_\\alpha |\\tau_{\\alpha+1}\n - \\tau_\\alpha|^{2}\n  \\end{split}\n\\end{equation*}\nThe last sum vanishes as the mesh of the partition shrinks to zero, thus\nthe claim follows.\n\\end{proof}\n\nIn this construction choosing the It\\^o version for the double integral\n$\\mathbb{X}$ was arbitrary. Alternatively we could have considered other\ndefinitions, e.g. Stratonovich integral and let\n$$\n(\\mathbb{X}_{\\text{Stra}})_{st}^{ij} = \\int_s^t \\int_s^u \\circ dX^i_v\n\\circ dX^j_u\n$$\nwhere $\\circ dX$ stands for Stratonovich (or symmetric) integration. By \nthe same procedure we find a regular version of $\\mathbb{X}_{\\text{Stra}}$ \nsuch that $\\|\\mathbb{X}_{\\text{Stra}}\\|_{2\\gamma} < \\infty$ and can \nconstruct the rough integral over the couple $(X,\\mathbb{X}_{\\text{Stra}})$ \nwhich we denote again $\\circ dX$. It is not difficult to prove that for\n$\\varphi \\in C^2$ it coincides with the familiar Stratonovich integral.\n\nThe relationship between the rough integrals based on the It\\^o and \nStratonovich constructions follows from the identity\n$$\n(\\mathbb{X}_{\\text{Stra}})_{st}^{ij} =\n(\\mathbb{X}_{\\text{It\\^o}})_{st}^{ij} + \\frac{1}{2} \\delta_{ij} (t-s)\n$$\nbetween It\\^o and Stratonovich stochastic iterated integrals. The\ncorrection is given by the increment of the function $t \\mapsto\n\\delta_{ij} t\/2$. Thus the two rough integrals are related by the \nfamiliar formula\n$$\n\\int_0^T \\varphi(X_u) \\circ dX_u = \\int_0^T \\varphi(X_u) dX_u +\n\\frac{1}{2} \\int_0^T \\text{div } \\varphi(X_u) du.\n$$\nThis is obtained directly from the definitions with the modified\nRiemann sums. \n\n\\begin{remark}\n\\rm{\nDue to the multiplicative property the possible choices for \n$\\mathbb{X}$ differ only by the increment of a function, i.e., \nif $\\mathbb{X}_1$ and $\\mathbb{X}_2$ both satisfy the \nmultiplicative property with respect to $X$, then\n$\n(\\mathbb{X}_1)_{st}^{ij}- (\\mathbb{X}_2)_{st}^{ij} = \nf^{ij}_t-f^{ij}_s\n$ \nwith a function $f \\in C([0,T],\\mathbb{R}^d\\times \\mathbb{R}^d)$.   \n}\n\\end{remark}\n\nFinally, it is not difficult to see the following regularity result.\n\\begin{corollary}\nThe map $\\varphi \\mapsto \\int_0^1 \\varphi(X_u) dX_u$ is continuous\nfrom $C^2(\\mathbb{R}^d,\\mathbb{R}^d)$ to $\\mathbb{R}$.  \n\\end{corollary}\n\n\\section{Stochastic currents}\n\\subsection{Lifting Wiener measure to the space of currents}\nLet $\\mathcal{X} = C(\\mathbb{R}, \\mathbb{R}^d)$ be path space, i.e., the space of continuous\nfunctions from $\\mathbb{R}$ to $\\mathbb{R}^d$. The $\\sigma$-field $\\mathop{\\mathrm{{\\cal F}}}\\nolimits$ is generated by \nthe coordinate process $\\mathcal{X} \\ni X \\mapsto X_t \\in \\mathbb{R}^d$. For $I \\subset \n\\mathbb{R}$ we denote by $\\mathop{\\mathrm{{\\cal F}}}\\nolimits_I$ the $\\sigma$-field over $\\mathcal{X}$ generated by the \nevaluations for points in $I$, and write $\\mathop{\\mathrm{{\\cal F}}}\\nolimits_T$ when $I =\n[-T,T]$. Also, we put for a shorthand $I^c = \\mathbb{R} \\setminus I$. \nFor $s \\leq t$, the set $\\{\\mathcal{F}_{st} = \n\\sigma(X_u : s \\le u \\le t)\\}_{s \\leq t}$ is the forward filtration \nstarting at $s \\in \\mathbb{R}$.\n\nDenote as before by $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$ Wiener measure defined on $\\mathcal{X}$ (instead of \n$C([0,\\infty),\\mathbb{R}^d)$ as more usual), and write $H_0 = -(1\/2)\\Delta$. \nFor any finite division $t_1 < t_2 < ... < t_n \\in \\mathbb{R}$  we have\n\\begin{equation} \\label{wiener}\n\\mathop{\\mathrm{{\\cal W}}}\\nolimits(F) = (f_1,e^{-(t_2-t_1)H_0} f_2 \\ldots \ne^{-(t_n-t_{n-1})H_0} f_n)_{L^2(\\mathbb{R}^d,dx)}\n\\end{equation}\nwhere $F = f_1(X_{t_1})\\cdots f_n(X_{t_n})$. Here it is understood\nthat $f_2, \\ldots, f_{n-1}$ act as multiplication operators for which \nwe use the same symbol as for the corresponding functions. The\noperator $e^{-tH_0}$ has the integral kernel \n\\begin{equation} \\label{(2.2)}\n\\Pi_t(x,y) = \\frac{1}{(2 \\pi t)^{d\/2}} \n\\exp \\left( - \\frac{1}{2t} (x-y)^2 \\right), \\quad x,y \\in \\mathbb{R}^d.\n\\end{equation}\nWe denote by $\\mathop{\\mathrm{{\\cal W}}}\\nolimits_I$ the measure $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$ restricted to $\\mathcal{X}_I = C(I,\\mathbb{R}^d)$.\nSimilarly, with given $T>0$, $\\xi, \\eta \\in \\mathbb{R}^d$  write \n\\begin{equation} \\label{(2.3)}\n\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{\\xi,\\eta}_I(F) = \\left( e^{-(-T-t_1)H_0} f_1 e^{-(t_2-t_1)H_0}f_2 \n\\ldots e^{-(t_n-t_{n-1})H_0} \\left( f_n e^{-(T-t_n)H_0}(\\cdot,\\eta) \n\\right) \\right) (\\xi)\n\\end{equation}\nwith $I=[-T,T]$ and let $\\mathop{\\mathrm{\\widehat{\\cal W}}}\\nolimits^{\\xi,\\eta}_I(F) = \\mathop{\\mathrm{{\\cal W}}}\\nolimits^{\\xi,\\eta}_I(F)\/ \n\\Pi_{2T}(\\xi,\\eta)$ be the Brownian bridge starting in $\\xi$ at $-T$ \nand ending in $\\eta$ at $T$. Under $\\mathop{\\mathrm{\\widehat{\\cal W}}}\\nolimits^{x,y}_{[-T,T]}$ the process \n$X_t$ is a Gaussian semimartingale (Brownian bridge) satisfying the SDE\n$$\ndX_u = - \\frac{y-X_u}{T-u} du + dB_u\n$$\nwhere $(B_u)_{u\\in[-T,T]}$ is a $\\mathop{\\mathrm{\\widehat{\\cal W}}}\\nolimits^{x,y}_{[-T,T]}$-Browian motion. \n\n\\medskip\nNext, let $\\D$ be a Banach space of functions from $\\mathbb{R} \\times \\mathbb{R}^d$\nto $\\mathbb{R}^d$ containing the space of smooth functions \n$C^\\infty(\\mathbb{R} \\times \\mathbb{R}^d; \\mathbb{R}^d)$. Denote with $\\|\\cdot\\|_\\D$ the\nBanach norm. Let $\\D'$ the topological dual of $\\D$, $\\Delta_I =\n\\{(s,t)\\in I^2 | s\\le t\\}$, and write $\\Delta = \\Delta_\\mathbb{R}$. \n\\begin{definition}\n$C \\in C(\\Delta ; \\D')$ is a \\emph{stochastic current} if it \nsatisfies the following properties:\n\\begin{enumerate}\n\\vspace{0.1cm}\n\\item[(1)]\n$\nC_{tt}(\\varphi) = 0, \\;\\; C_{su}(\\varphi) + C_{ut}(\\varphi) = \nC_{st}(\\varphi), \\;\\;\n$\nfor any $s\\le u\\le t$ and any $\\varphi \\in \\D$;\n\\vspace{0.1cm}\n\\item[(2)]\n\\emph{locality property:} \\quad $C_{st}(\\varphi)=0 \\;$ \nwhenever $\\;\\varphi(u,x)=0$ for all $u\\in[s,t]$, $x \\in \\mathbb{R}^d$.\n\\end{enumerate}\nWe denote $\\C \\subset C(\\Delta ; \\D')$ the space of stochastic\ncurrents. \n\\end{definition}\nSet $\\Xi = \\mathcal{X} \\times \\C$ endowed with the product topology and \nwith the Borel $\\sigma$-field (on whose component generated\nby $\\C$ we consider the topology of uniform convergence on bounded \nintervals). $\\Xi$ plays the role of joint path-current configuration \nspace. As a measurable space, it is endowed with a family of \n$\\sigma$-algebras \n$\\{\\A_{st}\\}_{t>s}$ such that $\\A_{st} = \\sigma(X_u, C_{uv}(\\varphi) \n: u,v \\in [t,s], \\varphi \\in \\D )$. Similarly, we can define the \nforward filtration $\\{\\A_{t}^+\\}_t = \\{\\A_{+\\infty,t}\\}_t$ and the \nbackward filtration $\\{\\A^-_t\\}_t = \\{\\A_{t,-\\infty}\\}_t$. The above \ndefinitions make sense also in the case that the parameter $t$ is \nrestricted to a bounded interval $I\\subset \\mathbb{R}$; in this case we denote \nwith $\\Xi_I$ the corresponding space. Whenever the limits make sense \nwe define $C^{+}_t(\\varphi) = \\lim_{s\\to\\infty} C_{ts}(\\varphi)$ and \n$C^-_t(\\varphi) = \\lim_{s\\to -\\infty}C_{st}(\\varphi)$.\n\n\n\n\n\\begin{definition}\n\\label{def:forward-current}\nA \\emph{forward current} (on $I\\subset \\mathbb{R}$) is a measure $\\eta$ on\n$\\Xi$ (on $\\Xi_I$) such  that the process $X$ is an \n$(\\eta,\\{\\mathop{\\mathrm{{\\cal F}}}\\nolimits_t^+\\}_t)$-semimartingale and\n\\begin{equation}\n  \\label{eq:curr-constr-0}\nC_{st}^X(\\varphi) = \\int_s^t \\varphi(u,X_u) dX_u, \n\\qquad \\eta-\\text{a.s.}  \n\\end{equation}\nfor any $(s,t)\\in \\Delta$ (or $\\Delta_I$) and any adapted $\\varphi \\in \n\\D$ where on the right hand side we have the standard It\\^o integral on \nthe semimartingale $X$. When $X$ is Brownian motion, we call $\\eta$ \n\\emph{(forward) Brownian current}. \n\\end{definition}\nIn order not to multiply terminology, unless confusion may arise we \nwill use the term \\emph{current} also for the elements of $\\D'$, of \n$\\Xi$, and for the laws on $\\Xi$ without making explicit distinction. \nFor unspecified bounded intervals $I$ we use the notation $C_I^X$ for\nthe associated current with integrator $X$.\n\nNext we want to construct a (non-trivial) measure on $\\Xi$ for the\nBrownian current. Thus we start from Wiener measure $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$ (similarly \nwe could have worked with the Brownian bridge $\\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^{x,y}$) and prove \nthat there exists a map $F: \\mathcal{X} \\ni \\omega \\mapsto F(\\omega)\\in\\C$ such\n that\n\\begin{equation}\n  \\label{eq:curr-constr-0-bis}\nF(\\cdot)_{st}(\\varphi) = \\int_s^t \\varphi(u,X_u) dX_u, \n\\qquad \\mathop{\\mathrm{{\\cal W}}}\\nolimits-\\text{a.s.}  \n\\end{equation}\nfor any adapted $\\varphi \\in \\D$, with the standard It\\^o integral \nat the right hand side. Then a measure $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp$ on $\\Xi$ can be \ndefined as the law of the couple $(X,F)$ under the measure $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$ and \nit will be a forward current. The existence of a regular version of \nmap $F$ is an interesting problem in itself which can be addressed \nby using the techniques developed in~\\cite{FGGT, FGR} for what \nconcerns the regular dependence on $\\varphi$. Unfortunately, the \ntopology $\\D$ which is implied by such approaches is unsuitable for \nour applications. Here we prefer to use the theory of rough paths \nwhich will provide the necessary regularity for the $F$ map in a \nmore convenient topology.\n\nIn Section ~\\ref{sec:rough-paths} we developed the basic tools of\nrough-path theory that we need in order to lift Wiener measure to \ncurrents. We do this next.\n\nFor any $\\alpha > 1$, let $\\D_\\alpha$ be the completion of the space of \nsmooth test vector fields with respect to the norm\n$$\n\\|\\varphi\\|_{\\D_\\alpha} = \\sup_{k \\in \\Z} (1+|k|)^\\alpha \n\\|\\varphi\\|_{\\rho,2,k,k+1}.\n$$ \nIn the following we will fix $\\alpha>1$ but otherwise arbitrarily small and \nwrite $\\D_\\alpha = \\D$.\n\\begin{lemma}\n\\label{lemma:rough}\nFor any $\\gamma \\ge 1\/3$, $\\rho > 1-\\gamma$, $t>s$ and $x,y\\in \\mathbb{R}^d$, \nthere exists a family of random variables $F \\in \\C$ such that\n(\\ref{eq:curr-constr-0}) holds with respect to $\\mathop{\\mathrm{{\\cal W}}}\\nolimits_{[s,t]}^{x,y}$, \nand which satisfy the pathwise bound\n\\begin{equation}\n  \\label{eq:F-bound}\n |F_{uv}(\\varphi)| \\le C_{\\gamma,\\rho,|t-s|} |u-v|^\\gamma (1+N_{[s,t]}(X))^3 \n \\|\\varphi\\|_{\\rho,2,s,t}\n\\end{equation}\nfor any $\\varphi \\in \\D$, $(u,v) \\in \\Delta_{[s,t]}$, where\n$N_{[s,t]}(X) = (\\|X\\|_{\\gamma,[s,t]} + \n\\|\\mathbb{X}^2\\|_{2\\gamma,[s,t]})$ and where\n$C_{\\gamma,\\rho,|t-s|}$ depends only on $\\gamma,\\rho,|t-s|$.\n\\end{lemma}\n\n\\begin{proof}\nDefine\n\\begin{equation*}\nF_{uv}(\\varphi) = \n\\lim_{\\delta\\tau_\\alpha \\to 0} \n\\sum_\\alpha \\left(\\varphi(\\tau_\\alpha,X_{\\tau_\\alpha}) \n+ \\nabla \\varphi(\\tau_\\alpha,X_{\\tau_\\alpha}) \n\\mathbb{X}^2_{\\tau_\\alpha,\\tau_{\\alpha+1}}\\right)\n\\end{equation*}\nwhere  $\\delta\\tau_\\alpha$ is the mesh of the partition and \n$\\mathbb{X}^2_{st}\\in C(\\mathbb{R}^2;\\mathbb{R}^d \\times \\mathbb{R}^d)$ is the \ntwice iterated It\\^o integral with respect to $X$. By the results \non rough path theory in Section \\ref{sec:rough-paths} (compare \nTheorem \\ref{th:rough-main}), this limit exists whenever \n$N_{[s,t]}(X)<\\infty$ and $\\|\\varphi\\|_{2,\\rho,s,t}<\\infty$. Otherwise \nset $F_{uv}(\\varphi) = 0$. Then $F$ is a well defined random variable \nobeying (\\ref{eq:F-bound}). Moreover,  $F_{su}(\\varphi)+F_{ut}\n(\\varphi)= F_{st}(\\varphi)$ and the locality property for $F$ holds\nby definition.\n\nBy straightforward estimates we can also prove that \n$\\mathbb{E}_{\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_I} [N_I(X)^3] < \\infty$ for any $x,y,I$. Using\nthis last result, the equivalence between $F$ and the It\\^o integral \nfor the adapted vector field $\\varphi$ can be proved by the same \napproach as the one used in the case of Wiener measure.\n\\end{proof}\n\n\nA direct consequence of Lemma~\\ref{lemma:rough} is that whenever \n$N_{[s,t]}(X) < \\infty$ the map $\\varphi \\mapsto F(\\omega)_{st}\n(\\varphi)$ can be considered as an element of $\\D'$. Moreover if \nwe let\n$$\n\\mathcal{N}_{\\alpha,p}(X) = \\sum_{k \\in \\Z} (1+|k|)^{-\\alpha} \nN_{[k,k+1]}(X)^p,\n$$\nthen whenever $\\mathcal{N}_{\\alpha,3}(X) < \\infty$, the boundary \ncurrents $C^+_t$ and $C^-_t$ are well defined for any $t$ as elements \nof $\\D'$.\n\n\\begin{lemma}\n\\label{lemma:W-lift}\nFor every bounded $I \\subset \\mathbb{R} $ there exists a unique Brownian\ncurrent $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{\\sharp,x,y}_I$ on $\\Xi_I$. A similar statement holds for\nthe measures $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp$ with first marginal $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$. Moreover, since\nunder $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$ we have  $\\mathcal{N}_{\\alpha,3}(X) < \\infty$ a.s., the\nboundary currents are well defined under $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp$.\n\\end{lemma}\n\\begin{proof}\nThe existence of the lifted measure for Brownian bridge $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_I$ \nis essentially contained in Lemma~\\ref{lemma:rough}. Its uniqueness \nis a direct consequence of the property~(\\ref{eq:curr-constr-0}). \nThe proof in the case of $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$ is similar and we are left to prove that\nunder $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp$ the boundary currents are well defined. The integrals \nin every interval $[i,i+1]$ are well defined, moreover\n$$\n\\left|\\int_{i}^{i+1}  \\varphi(u,X_u) dX_u\\right|\\le (1+|i|)^{-\\alpha} \n\\|\\varphi\\|_{\\mathcal D} [1+N_{[i,i+1]}(X)]^3  \n$$\nso that the series\n$\n\\int_{t}^{+\\infty}\\varphi(u,X_u) dX_u =   \n\\sum_{i>t-1}\\int_{i\\vee t}^{i+1}  \\varphi(u,X_u) dX_u\n$\nis absolutely convergent if $\\mathcal{N}_{\\alpha,3}(X) < \\infty$.\nUnder $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$ we have\n$\n\\mathbb{E}_{\\mathop{\\mathrm{{\\cal W}}}\\nolimits} [(1+N_{[i,i+1]}(X))^3|X_0=x] \\le C\n$ \nuniformly in $i \\in \\Z$ and $x\\in \\mathbb{R}^d$ thus\n$$\n\\mathbb{E}_{\\mathop{\\mathrm{{\\cal W}}}\\nolimits} [\\mathcal{N}_{\\alpha,3}(X)|X_0=x]  = \n\\sum_{i \\in \\Z}(1+|i|)^{-\\alpha} \n\\mathbb{E}_{\\mathop{\\mathrm{{\\cal W}}}\\nolimits} [N_{[i,i+1]}(X)^3|X_0=x] \n\\le \nC  \\sum_{i \\in \\Z} (1+|i|)^{-\\alpha} < \\infty\n$$\nas soon as $\\alpha > 1$. This implies that $\\mathcal{N}_{\\alpha,3}(X)$\nis $\\mathop{\\mathrm{{\\cal W}}}\\nolimits$-a.s. finite.\n\\end{proof}\n\n\n\\begin{remark}\n\\rm{\nNote that the lifting of a measure from $\\mathcal{X}$ to $\\Xi$ is in general \nnot unique. For instance, we could decide to add some other term to \nthe definition of the current,\n$$\n\\tilde C_{st}^X(\\varphi) = C_{st}^X(\\varphi) + \\int_s^t \\mathrm{div}\\,\n\\varphi(t, X_t) dt\n$$\nand obtain a different lifted measure (which is no longer a forward \ncurrent). It would be interesting to explore whether different lifts \nmay have a different physical meaning in the models. For instance, \nnon-relativistic particles with spin can be (partially) described by \na current $C^\\sigma$ defined as\n$$\nC^X_{st}(\\varphi)[\\sigma] = C_{st}^X(\\varphi) + \\int_s^t \\sigma_t \n\\cdot \\mathrm{curl}\\, \\varphi(t, X_t) dt\n$$\nwhere $(\\sigma_t)_t$ is a vector-valued Poisson process describing\nthe spin of the particle~\\cite{Jona,Jona2,HL1}.  \n}\n\\end{remark}\n\n\n\\subsection{It\\^o current}\n\\label{sec:ito-current}\nBeside defining Brownian currents, it will be useful for our purposes \nbelow to define currents for Brownian paths subjected to a potential \n(or penalty function) $V$. The reason is that we need a sufficiently\nstrong confining mechanism of paths in order to investigate the effect\nof a pair interaction on them (given by double stochastic integrals). \nThe translation invariant ($V \\equiv 0$) regime is presently little\nunderstood. \n\nWhile we properly introduce potentials only in the next section, we \nrequire here that the integral kernel\n\\begin{equation} \n\\label{fkf}\n\\hat\\pi_{t-s}(x,y) = \\int e^{- \\int_s^t V(X_u)du} \\, \nd\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_{[s,t]}(X), \\quad \\forall s<t \\in \\mathbb{R}, \\;\n\\forall x,y \\in \\mathbb{R}^d\n\\end{equation}\nexists. With assumptions on $V$ listed in Section 4.1 below this can be\nensured, and then furthermore the map $(t,x,y) \\mapsto \\hat \\pi_t(x,y)$ \nis jointly continuous and bounded on $(0,\\infty) \\times \\mathbb{R}^d \\times \n\\mathbb{R}^d$ giving rise to a semigroup $S_t$ via the formula\n$$\n(S_t f) (x) = \\int \\hat\\pi_{t}(x,y) f(y).\n$$\nMoreover, for every $t>0$ the semigroup $S_t$ is a bounded operator \nfrom $L^p$ to $L^q$ for every $ 1 \\leq p \\leq q \\leq \\infty$, and by \nthe Feynman-Kac formula and the Hille-Yoshida and Stone Theorems it\ncan be written as $S_t=e^{-tH}$, with $H$ coinciding with the\nSchr\\\"odinger operator $H_0+V$ on $C_0^\\infty(\\mathbb{R}^d)$. In addition, \n$e^{-tH}f$ is a continuous function for every $f \\in L^p, p \\in \n[1,\\infty]$, $\\forall t > 0$. \n\nLet $\\Psi$ be a ground state of the Schr\\\"odinger operator $H$, i.e., \na normalized eigenfunction in $L^2(\\mathbb{R}^d,dx)$ lying at $\\inf\\mathop{\\mathrm{Spec}}\\nolimits H$. \nUnder the conditions given in Section 4.1. this ground state is unique \nand has a strictly positive version. Using this we define the\nprobability measure $\\nu$ on $(\\mathcal{X},\\mathop{\\mathrm{{\\cal F}}}\\nolimits,\\mathop{\\mathrm{{\\cal W}}}\\nolimits)$ by\n\\begin{equation} \n\\label{pphi1}\n\\nu(A) = Z_T^{-1} \\int  dx \\, \\Psi(x) \\int dy \\, \\Psi(y) \\int 1_{A}(X) \ne^{-\\int_{-T}^T(X_t)dt} \\, d\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_T(X)\n\\end{equation}\nwith normalizing constant $Z_T$, $A \\in \\mathop{\\mathrm{{\\cal F}}}\\nolimits_{T}$. This measure can be\nextended to a measure $\\nu$ on the full $\\mathop{\\mathrm{{\\cal F}}}\\nolimits$ by making use of the \nfacts $e^{-tH}\\Psi = \\Psi$ and $\\norm[2]{\\Psi} = 1$. The Feynman-Kac \nformula and the Markov property of Brownian motion imply that \n$\\{\\nu_T\\}$ given on $\\mathop{\\mathrm{{\\cal F}}}\\nolimits_{T}, \\,  T>0$, define a consistent family of \nprobability measures. In particular, $\\nu$ satisfies the DLR equations \nand thus is a Gibbs measure relative to Brownian motion for potential \n$V$; for further details see \\cite{BL}. \n\nMoreover, $\\nu$ is the law of a reversible diffusion process with \nstationary distribution $d\\omega = \\Psi^2 \\, dx$ and stochastic \ngenerator $H_{\\omega}$ acting in $L^2(\\mathbb{R}^d,d\\omega)$ as\n$$ \nH_{\\omega} f = \\frac{1}{\\Psi} H (\\Psi f) = H_0 f - \n\\left(\\nabla \\log \\Psi, {\\nabla f}\\right)_{\\mathbb{R}^d}.\n$$\nThis process is called \\emph{It\\^o diffusion} (or \n\\emph{$P(\\phi)_{1}$-process} in quantum field theory). Its transition \nprobabilities are given by \n\\begin{equation} \\label{transition prob}\n\\nu ( f(X_{t+s}) | X_{s} = x) = \\int \\pi_{t}(x,y) f(y) \\, d\\omega(y),\n\\end{equation}\nwhere\n\\begin{equation} \n\\pi_{t}(x,y) = \\frac{\\hat\\pi_{t}(x,y)}{\\Psi(x)\\Psi(y)}\n\\label{pito}\n\\end{equation}\nis the transition density of $\\nu$ with respect to its stationary \ndistribution. The It\\^o process is Markovian, reversible, and has a \nversion with continuous paths. Moreover, it is a Brownian\nsemi-martingale with respect to either the forward or the backward \nfiltration, in particular it is the stationary solution of the forward \nstochastic differential equation \n\\begin{equation}\n  \\label{eq:diffusion-eq}\ndX_t = \\nabla \\log \\Psi(X_t) dt + dB_t, \n\\end{equation}\nwhere $B_t$ is Brownian motion with respect to the forward filtration.\n\nFor the It\\^o bridge, i.e., the regular conditional probability \n$\\nu^{x,y}_T$ of $\\nu$ given $X_{-T} = x$, $X_{T} = y$, we will use\nthe following representation. Take (\\ref{pphi1}) describing the density \nof the measure $\\nu_T$ with respect to Brownian motion. Then\n\\begin{equation}\n  \\label{eq:bridge-2}\n\\nu_T^{x,y}(A) = \\frac{ \\Pi_{2T}(x,y)}{Z_T\\, \\Psi(x) \\Psi(y) \\pi_{2T}(x,y)}\n\\mathbb{E}_{\\mathop{\\mathrm{\\widehat{\\cal W}}}\\nolimits^{x,y}_{[-T,T]}} \\left[ 1_A(X) e^{- V_{[-T,T]}(X) }\\right],  \n\\end{equation}\nwith $\\Pi_t$ the Brownian transition kernel (\\ref{(2.2)}). This\nformula can be checked by noting that \n\\begin{equation*}\n  \\begin{split}\n&\n\\hat \\mathbb{E}_{\\nu_T} \\left[f(x) g(y) \\mathbb{E}_{\\nu_T^{x,y}}[Q] \\right] \\\\ & \n\\quad \n= Z_T^{-1} \\hat \\mathbb{E}_{\\mathcal{W}_{[-T,T]}} \\left[f(x)g(y)\\Psi(x)\\Psi(y) \n\\mathbb{E}_{\\mathop{\\mathrm{\\widehat{\\cal W}}}\\nolimits_{[-T,T]}^{x,y}}[Q(X) e^{- \\int_{-T}^TV(X_t)dt}] \\right] \\\\ & \n\\quad \n= Z_T^{-1} \\mathbb{E}_{\\mathcal{W}_{[-T,T]}} \\left[\\Psi(X_{-T})\\Psi(X_T) \nf(X_T) g(X_{-T}) Q(X) e^{-\\int_{-T}^T V(X_t)dt}\\right] \\\\ & \n\\quad \n= \\mathbb{E}_{\\nu_T} \\left[f(X_T) g(X_{-T})Q(X) \\right]\n\\end{split}\n\\end{equation*}\nwhere $\\hat \\mathbb{E}$ denotes the expectation in a new probability\nspace whose coordinate process is denoted $\\hat X$, and with $\\hat\nX_{-T} = x$, $\\hat X_T = y$. \n\nThe assumption that $V$ is Kato-class (see Section 4.1) implies that\nthe It\\^o bridge measure $\\nu^{x,y}_{I}$ is absolutely continuous with\nrespect to the Brownian bridge measure $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_{I}$. This allows us \nto use the lifting result proved in Lemma~\\ref{lemma:W-lift} to show \nthat also $\\nu^{x,y}_{I}$ allows a lift to the space of currents. \n\\begin{lemma}\n\\label{lemma:nu-lift}\nFor every bounded $I \\subset \\mathbb{R}$ there exists a forward current \n$\\nu^{\\sharp,x,y}_I$ on $\\Xi_I$ such that is first marginal is\n$\\nu^{x,y}_I$. A similar statement holds for the stationary measures \n$\\nu^\\sharp$ with first marginal $\\nu$. Moverover, since under $\\nu$ \nwe have  $\\mathcal{N}_{\\alpha,3}(X) < \\infty$ a.s., the boundary \ncurrents are well defined under $\\nu^\\sharp$.\n\\end{lemma}\n\\begin{proof}\nWe have\n$$\n\\mathbb{E}_{\\nu^{x,y}_I} [N_I(X)^3] \\le \nC \\left(\\mathbb{E}_{\\widehat \\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x,y}_I} [N_I(X)^6]\\right)^{1\/2}  \n\\left(\\mathbb{E}_{\\nu^{x,y}_I} [e^{-2\\int_I V(X_t)dt}]\\right)^{1\/2}  \n\\le C.\n$$\nHence the map $F$ defined in Lemma~\\ref{lemma:rough} is well defined\nand coincides almost surely with the It\\^o integral. This allows to\nconstruct the lifted measures $\\nu^{\\sharp,x,y}_I$ and $\\nu^{\\sharp}$. \nMoreover, by using stationarity of $\\nu$ we have that\n$\n\\mathbb{E}_{\\nu} [N_{[i,i+1]}(X)^3] \\le C\n$\nuniformly in $i\\in \\Z$, thus the boundary currents are well defined \nunder $\\nu^\\sharp$.\n\\end{proof}\n\n\n\\section{Gibbs measures on Brownian currents}\n\\subsection{Conditions on the potentials}\n\\label{sec:potentials}\nWe use the same terminology of the usual DLR theory and introduce \n``potentials\" and ``energy functionals\" below. \n\nAn \\emph{external potential} is a Lebesgue measurable function $V: \n\\mathbb{R}^d \\to \\mathbb{R}$ that we will choose from the Kato class, i.e., an element \nof the space $\\mathcal{K}(\\mathbb{R}^d)$ defined by the condition\n\\begin{equation}\n\\lim_{r \\to 0} \\sup_{x \\in \\mathbb{R}^d} \\int_{B_r(x)} \n|g(y-x)V(y)| \\, dy = 0,\n\\end{equation}\nwith $B_r(x)$ the ball of radius $r$ centered at $x$, and \n\\begin{equation}\n g(x) = \\left\\{ %\n\\begin{array}{ll} \n        |x| & \\mbox{if} \\;\\; d = 1 \\\\         \n        -\\ln|x| & \\mbox{if} \\;\\; d = 2 \\label{k2}\\\\\n        |x|^{2-d} & \\mbox{if}   \\;\\;  d \\geq 3.\n    \\end{array} \\right. \n\\end{equation}\nThis space is large enough to contain many choices of interest,\nwhile allowing the Feynman-Kac formula for the Schr\\\"odinger\nsemigroup $e^{-tH}$, $t \\geq 0$, to hold. This is generated by  \nthe Schr\\\"odinger operator $H = H_0 + V$ defined on \n$L^2(\\mathbb{R}^d,dx)$ as a form sum ($V$ regarded as a multiplication \noperator). For Kato-class potentials $H$ is essentially \nself-adjoint on the form core $C^\\infty_0(\\mathbb{R}^d)$. In addition, \nwe will require of $V$ to be such that \n\\begin{enumerate}\n\\item[(1)]\n$H$ has a unique strictly positive eigenfunction (ground state) \n$\\Psi$ at $E = \\inf\\mathop{\\mathrm{Spec}}\\nolimits H$, with the property that $\\Psi \\in\nL^1 \\cup L^\\infty$;\n\\item[(2)] \n$e^{-tH}$ is intrinsically ultracontractive.\n\\end{enumerate}\nRecall the meaning of the latter property. Write $d\\omega = \\Psi^2 \ndx$ on $\\mathbb{R}^d$ as before, and define the isometry (\\emph{ground \nstate transform}) $j: L^2(\\mathbb{R}^d,d\\omega) \\rightarrow L^2(\\mathbb{R}^d,dx)$, \n$f \\mapsto \\Psi f$. Then $D(H_\\omega) = j^{-1}D(H)$ and $H_\\omega \nf=(j^{-1}Hj)f = (1\/\\Psi) H(\\Psi f) = -(1\/2)\\Delta f-({\\nabla\\ln\\Psi},\n\\nabla f)_{\\mathbb{R}^d}$, for every $f \\in D(H_\\omega)$. The associated \nsemigroup $e^{-tH_\\omega}$ exists for all $f\\in L^2(\\mathbb{R}^d,d\\omega)$ \nand $t \\geq 0$. $e^{-tH}$ is \\emph{intrinsically ultracontractive} \nwhen $e^{-tH_\\omega}$ is \\emph{ultracontractive}, i.e., it maps \n$L^2(\\mathbb{R}^d, d\\omega)$ into $L^\\infty(\\mathbb{R}^d,d\\omega)$ continuously. \nEquivalently, this means that $\\|{e^{-tH_\\omega}}\\|_{2,\\infty} < \n\\infty$, $\\forall t> 0$, and it is a monotonically decreasing \nfunction in $t$. Moreover, the integral kernel (\\ref{pito}) of \n$e^{-tH_\\omega}$ satisfies $0 \\leq \\pi_t(x,y) \\leq \n\\|e^{-(t\/2)H_\\omega}\\|_{2,\\infty}^2$ almost surely.\n\nThese conditions are in particular satisfied for $V$ bounded \nfrom below, continuous, and sufficiently confining, i.e., for \nwhich there exist constants $C_1, C_2 > 0$, $C_3, C_4\\in \\mathbb{R}$, \nand $a,b$ with $2 < a < b < 2a - 2$ such that the positive \npart of the potential, $V^+ = \\sup \\{0,V\\}$ satisfies\n\\begin{equation} \\label{iuccond}\nC_1|x|^a + C_3 \\leq V^{+}(x) \\leq C_2|x|^b + C_4.\n\\end{equation}\n\n\n\\bigskip\n\nA \\emph{pair interaction potential} is a Lebesgue measurable \nfunction $W: \\mathbb{R}^d \\times \\mathbb{R} \\to \\mathbb{R}$, even in both of its \nvariables, which we require to \n\\begin{enumerate}\n\\item[(1)]\nhave positive Fourier transform;\n\\item[(2)]\nsatisfy the regularity condition that there exists \n$M_{I,\\beta} \\in \\mathbb{R}$ such that\n\\begin{equation}\n\\sup_{x\\in \\mathbb{R}^d ,t\\in I} \\|W(x,t)\\|_{\\D_{\\beta}} \\leq\nM_{I,\\beta},\n\\label{a22}\n\\end{equation}\nfor $\\beta > \\max\\{\\alpha, 3\\}$ and every bounded $I \\subset \n\\mathbb{R}$.\n\\end{enumerate}\nThe requirement $\\beta>\\alpha$ is needed for having a well \ndefined interaction energy (actually in our applications for \nany $\\beta>1$ there is a suitable $\\alpha$ so that this holds), \nwhile $\\beta>3$ is a decay condition sufficient for ensuring \nthe convergence of the cluster expansion in Chapter 5 below. \n \nAn example satisfying these conditions is $W^\\rho$ seen in \nSection 1.\n\n\\bigskip\nFinally we write down the energies appearing in the definition of \nthe densities of Gibbs measures we are going to study. With given \npair potential $W$, for all $a,b \\in \\mathcal{D}'$ consider the \nfunction $\\psi_{k,\\varpi}(x,t) = e^{ik \\cdot x +i\\varpi t}\\in \n\\mathcal{D}$ and define the (possibly unbounded) quadratic form \n$$\n\\langle a,b \\rangle_W := \\int \\widehat W(k,\\varpi) \na(\\psi_{k,\\varpi})b(\\overline{\\psi_{k,\\varpi}}) dk d\\varpi. \n$$\nBy using the quadratic form, \nfor all bounded $I \\subset \\mathbb{R}$ and every $\\mathbb{X} = (X, C^X), \n\\, \\mathbb{Y} = (Y, C^Y) \\in \\Xi$ define the \\emph{internal energy \nfunctional}\n\\begin{equation}\nH_I(\\mathbb{X}) = V_I(X)  + \\frac{\\lambda }{2}\\langle \nC^X_I, C^X_I \\rangle_W,\n\\label{hin}\n\\end{equation}\nand \n\\emph{interaction functional}\n\\begin{equation}\nH_I(\\mathbb{X}|\\mathbb{Y}) =  V_I(X) + \n\\frac{\\lambda}{2}\\langle C^X_I, C^X_I \\rangle_W + \n\\lambda  \\langle C^X_I, C^Y_{I^c} \\rangle_W,\n\\label{hint}\n\\end{equation}\nwith parameter $\\lambda \\in \\mathbb{R}$, where we wrote $V_I(X) = \n\\int_I V(X_t) dt$.\n\n\n\n\n\n\n\\subsection{Gibbs specifications}\n\n\nOn $\\Xi$ with its associated $\\sigma$-Borel field $\\A$ we take now \n$\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp$ as reference measure and define a Gibbs specification.\n\\begin{definition}[Gibbs specification]\nTake the regular version $\\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^\\sharp(d\\mathbb{X}|\\mathbb{Y})$ of \nthe measure $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp$ conditional on $\\mathbb{Y}$ in $\\A_{I^c}$, \nand $H_I(\\mathbb{X})$ given by (\\ref{hin}). We call the family of\nprobability kernels $\\{\\mu^\\sharp_I\\}_I$ on $\\Xi$ indexed by the \nbounded intervals $I \\subset \\mathbb{R}$, \n\\begin{equation}\n\\label{eq:curr-finite-vol}\n\\mu^\\sharp_I(d\\mathbb{X}) = \\frac{e^{-H_I(\\mathbb{X})}}\n{Z_I} \\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp_I(d\\mathbb{X})  \n\\end{equation}\na \\emph{Gibbs specification on Brownian currents with free \nboundary condition}. Take $H_I(\\mathbb{X}|\\mathbb{Y})$ given \nby (\\ref{hint}). We call the family $\\{\\rho^\\sharp_I\\}_I$ on\n$\\Xi$,\n\\begin{equation}\n\\label{eq:spec}\n\\rho^\\sharp_I(d\\mathbb{X}|\\mathbb{Y}) = \n\\frac{e^{-H_I(\\mathbb{X}|\\mathbb{Y})}}{Z_I(\\mathbb{Y})} \n\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp_I(d\\mathbb{X} | \\mathbb{Y})  \n\\end{equation}\na \\emph{Gibbs specification on Brownian currents with boundary\ncondition $\\mathbb{Y}$}. \n\\end{definition}\n\n\\begin{definition}[Gibbs measure]\nA probability measure $\\mu$ on $(\\Xi,\\A,\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp)$ is a \n\\emph{Gibbs measure} for the potentials $V$ and $W$ if it is \nconsistent with the specification $\\{\\rho^\\sharp_I\\}_I$, i.e.,\nthere exists a version of its conditional probabilities with\nrespect to the family $\\{\\A_{I^c}\\}_I$ which agrees with \n$\\{\\rho^\\sharp_I\\}_I$ for all bounded $I \\subset \\mathbb{R}$. \n\\end{definition}\nIn the following chapter our main concern will be to prove the\nexistence of such Gibbs measures. \n\nOn Gibbs specifications here is a first result. \n\\begin{lemma}\nThe family $\\{ \\rho_I^\\sharp \\}_I$ is consistent, i.e., for \nevery pair of bounded intervals $I \\subset J \\subset \\mathbb{R}$ \nwe have $\\int\\int F(\\mathbb{X})\\rho_I^\\sharp(d\\mathbb{X}|\\mathbb{Y})\n\\rho_J^\\sharp(d\\mathbb{Y}|\\mathbb{Z}) = \\int  F(\\mathbb{X})\n\\rho_J^\\sharp(d\\mathbb{X}|\\mathbb{Z})$, for any bounded measurable $F:\n\\Xi \\to \\mathbb{R}$.\n\\end{lemma}\n\\begin{proof}\nThe family  $\\{ \\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^\\sharp \\}_I$ is consistent by its definition. \nHence\n\\begin{equation*}\n  \\begin{split}\n\\int \\int & F(\\mathbb{X})\n\\rho_I^\\sharp(d\\mathbb{X}|\\mathbb{Y})\n\\rho_J^\\sharp(d\\mathbb{Y}|\\mathbb{Z}) \n\\\\ & =\n     \\int \\int F(\\mathbb{X}) \\frac{e^{-V_I(X)-(\\lambda\/2)\n     \\langle C^X_I, C^X_I \\rangle_W - \n     \\lambda\\langle C^X_I,C^Y_{I^c}\\rangle_W}}{Z_I(\\mathbb{Y})}\n\\\\ & \\quad \\quad \\times\n \\frac{e^{-V_J(Y)-(\\lambda\/2)\\langle C^Y_J, C^Y_J \\rangle_W-\n \\lambda\\langle C^Y_J,C^Z_{J^c} \\rangle_W}}{Z_J(\\mathbb{Z})}\n\\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^\\sharp(d\\mathbb{X}|\\mathbb{Y})\\mathop{\\mathrm{{\\cal W}}}\\nolimits_J^\\sharp(d\\mathbb{Y}|\\mathbb{Z}) \n\\\\ &=  \n     \\int \\int F(\\mathbb{X}) \\frac{e^{-V_I(X)-(\\lambda\/2)\n     \\langle C^X_I, C^X_I \\rangle_W-\n     \\lambda\\langle C^X_I,C^Y_{I^c}\\rangle_W}}{Z_I(\\mathbb{Y})}\n\\\\ & \\quad \\quad \\times\n{e^{-V_I(Y)-(\\lambda\/2)\\langle C^Y_I, C^Y_I \\rangle_W\n-\\lambda \\langle C^Y_I,C^Z_{J^c} \\rangle_W -\n\\lambda\\langle C^Y_I,C^Y_{K} \\rangle_W }}{}\n\\\\ & \\quad \\quad \\times\n \\frac{e^{-V_K(Y)-(\\lambda\/2)\\langle C^Y_K, C^Y_K \\rangle_W-\n \\lambda\\langle C^Y_K,C^Z_{J^c} \\rangle_W}}{Z_J(\\mathbb{Z})}\n\\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^\\sharp(d\\mathbb{X}|\\mathbb{Y})\\mathop{\\mathrm{{\\cal W}}}\\nolimits_J^\\sharp(d\\mathbb{Y}|\\mathbb{Z}) \n  \\end{split}\n\\end{equation*}\nwhere we split off $C^Y_J = C^Y_I + C^Y_K$ with $K = J\\backslash \nI$. This gives for the right hand side\n\\begin{equation*}\n  \\begin{split}\n\\quad  &\n     \\int \\int F(\\mathbb{X}) \\frac{e^{-V_I(X)-(\\lambda\/2)\n     \\langle C^X_I,C^X_I \\rangle_W-\n     \\lambda\\langle C^X_I,C^Y_{I^c}\\rangle_W}}{Z_I(\\mathbb{Y}_2)}\n\\\\ & \\quad \\quad \\times\n\\left( \\int {e^{-V_I(Y_1)-(\\lambda\/2)\\langle C^{Y_1}_I,C^{Y_1}_I\n\\rangle_W-\\lambda\\langle C^{Y_1}_I,C^Z_{J^c} \\rangle_W -\n\\lambda\\langle C^{Y_1}_I,C^{Y_2}_{K} \\rangle_W }}{} \n \\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^\\sharp(d\\mathbb{Y}_1|\\mathbb{Y}_2)\\right)\n\\\\ & \\quad \\quad \\times\n \\frac\n{e^{-V_K(Y_2)-(\\lambda\/2)\\langle C^{Y_2}_K, C^{Y_2}_K \\rangle_W-\n\\lambda\\langle C^{Y_2}_K,C^Z_{J^c} \\rangle_W}}{Z_J(\\mathbb{Z})}\n\\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^\\sharp(d\\mathbb{X}|\\mathbb{Y}_2) \\mathop{\\mathrm{{\\cal W}}}\\nolimits_K^\\sharp(d\\mathbb{Y}_2|\\mathbb{Z}), \n  \\end{split}\n\\end{equation*}\nwhere we used the fact that \n$\\int \\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^\\sharp(d\\mathbb{Y}_1|\\mathbb{Y}_2)\\mathop{\\mathrm{{\\cal W}}}\\nolimits_K^\\sharp\n(d\\mathbb{Y}_2|\\mathbb{Z}) = \\mathop{\\mathrm{{\\cal W}}}\\nolimits_J^\\sharp(d\\mathbb{Y}_1|\\mathbb{Z})$. Note that \nthe expression between the brackets equals $Z_I(\\mathbb{Y}_2)$, \nthus we further obtain\n\\begin{equation*}\n  \\begin{split}\n\\quad & \n     \\int \\int F(\\mathbb{X}) e^{-V_I(X)-(\\lambda\/2)\n     \\langle C^X_I, C^X_I \\rangle_W-\n     \\lambda\\langle C^X_I,C^Y_{I^c} \\rangle_W}\n\\\\ & \\quad \\quad \\times\n \\frac\n{e^{-V_K(Y_2)-(\\lambda\/2)\\langle C^{Y_2}_K, C^{Y_2}_K \\rangle_W -\n\\lambda\\langle C^{Y_2}_K,C^Z_{J^c} \\rangle_W}}{Z_J(\\mathbb{Z})}\n\\mathop{\\mathrm{{\\cal W}}}\\nolimits_I^\\sharp(d\\mathbb{X}|\\mathbb{Y}_2) \\mathop{\\mathrm{{\\cal W}}}\\nolimits_K^\\sharp(d\\mathbb{Y}_2|\\mathbb{Z}) \n\\\\ \\quad &=  \n     \\int  F(\\mathbb{Y})\n\\frac{e^{-V_J(Y)-(\\lambda\/2)\\langle C^{Y}_J, C^{Y}_J \\rangle_W -\n\\lambda\\langle C^{Y}_J,C^Z_{J^c} \\rangle_W}}{Z_J(\\mathbb{Z})}\n\\mathop{\\mathrm{{\\cal W}}}\\nolimits_J^\\sharp(d\\mathbb{Y}|\\mathbb{Z}) = \\int F(\\mathbb{Y}) \n\\rho_J(d\\mathbb{Y}|\\mathbb{Z}).\n  \\end{split}\n\\end{equation*}\n\\end{proof}\n\n\nThe forward current $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp$ has the key property that\n$\nC^X_{st}(\\varphi) = \\int_s^t \\varphi(u,X_u) dX_u\n$,  \n$\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp$-a.s. for all $(s,t)\\in \\Delta$ and all adapted \n$\\varphi\\in \\D$. This will enable us to show that the finite \nvolume measures $\\mu_I$ coincide with the marginals of the \nmeasures $\\mu^\\sharp_I$ on the first component of the product \n$\\Xi$. The specification $\\{\\rho^\\sharp_I\\}$ can then be \nconsidered as a suitable rigorous replacement for the DLR \ndescription of the infinite-volume limit. A Gibbs measure on \n$\\mathcal{X}$ will then be a measure for which there exists a unique \nlift to the space $\\Xi$ of currents satisfying the relation~\n(\\ref{eq:curr-constr-0}) ensuring the identification of the \ncurrent with the stochastic integral and which satisfy the \nDLR conditions with respect to the specification \n$\\{\\rho^\\sharp_I\\}$.\n\nTo show that the specification is well defined we rewrite the \nvarious terms using the fact that, under the measure \n$\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp_I(d\\mathbb{X}|\\mathbb{Y})$ we have pathwise \nequality between the current $C^X$ and the stochastic integral \nwith respect to $X$ for adapted integrands belonging to $\\D$. \nThen\n\\begin{equation*}\n  \\begin{split}\n\\langle C_I^X, C_I^X \\rangle_W  & = \\int \\widehat W(k,\\varpi)\nC_I^X(\\psi_{k,\\varpi}) C_I^X(\\overline{\\psi_{k,\\varpi}})\ndk d\\varpi\n   \\\\ &   = \\int \\widehat W(k,\\varpi) |C_I^X(\\psi_{k,\\varpi})|^2\n             dk d\\varpi \n   \\\\ &   = \\int \\widehat W(k,\\varpi)\n\\left|\\int_I \\psi_{k,\\varpi}(t,X_t) dX_t \\right|^2 \ndk d\\varpi = 2 W_I(X)\n  \\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n  \\begin{split}\n\\langle C_I^X, C_{I^c}^Y \\rangle_W & = \n\\int \\widehat W(k,\\varpi) C_I^X(\\psi_{k,\\varpi}) C_{I^c}^Y\n(\\overline{\\psi_{k,\\varpi}}) dk d\\varpi \n\\\\\n & =\nC_I^X\\left( \\int \\widehat W(k,\\varpi)\\psi_{k,\\varpi} \nC_{I^c}^Y(\\overline{\\psi_{k,\\varpi}}) dk d\\varpi \\right)\n\\\\\n & =\nC_I^X\\left( w^{C_{I^c}^Y}\\right) = \\int_{-T}^{T}\nC_{I^c}^Y(W(\\cdot - X_s,\\cdot - s)) dX_s,\n  \\end{split}\n\\end{equation*}\nwith\n$$\nw^C(x,t) = \\int \\widehat W(k,\\varpi)\\psi_{k,\\varpi}(t,x)\nC(\\overline{\\psi_{k,\\varpi}}) dk d\\varpi  = \nC(W(x-\\cdot,t-\\cdot)).\n$$\nBy using these equalities it is seen that the \nspecification~(\\ref{eq:spec}) takes the form\n\\begin{equation}\n\\label{eq:spec2}\n\\rho^\\sharp_I(d\\mathbb{X}|\\mathbb{Y}) =\n\\frac{e^{-V_I(X)-\\lambda W_I(X)-\n\\lambda \\int_I w^{C_{I^c}^Y}(u,X_u)dX_u}}{Z_I(\\mathbb{Y})} \n\\mathop{\\mathrm{{\\cal W}}}\\nolimits^\\sharp_I(d\\mathbb{X} | \\mathbb{Y})  \n\\end{equation}\nand it is well defined as soon as the exponential weight is integrable\nand the integral is different from zero. The conditions on $V$ and $W$ \nmake sure this is true. Indeed, for Kato-class potentials exponential \nintegrability is a consequence of Khasminskii's Lemma \\cite{Sim82}. \nMoreover, since the Fourier transform of $W$ is positive by assumption \nand $\\lambda >0$, the internal energy term is negative and thus \nexponentially integrable without any further restriction. For the \ninteraction with the boundary current we have\n$$\n|w^{C_{I^c}^Y}(x,t)| = |C_{I^c}^Y(W(x,t))| \\le M_I \\|C_{I^c}^Y\\|_{\\D'} \n$$\nwith $M_I = \\sup_{x\\in \\mathbb{R}^d,t\\in I} \\|W(x,t)\\|_{\\D}$, which by\ncondition (2) on $W$ is finite. Hence the stochastic integral in the \nexponent has a bounded and adapted integrand and thus by standard \ntechniques it follows that it is exponentially integrable for any\nvalue of $\\lambda$.\n\n\n\\medskip\nBy making use of the It\\^o current defined in\nSection~\\ref{sec:ito-current}, the specification $\\{\\rho^\\sharp_I\\}_I$ \ncan be finally written as\n\\begin{equation}\n\\label{eq:specc2}\n\\rho^\\sharp_I(d\\mathbb{X}|\\mathbb{Y}) =\n\\frac{e^{-\\lambda W_I(X)-\n\\lambda \\int_{-T}^T w^{C_{I^c}^Y}(u,X_u)dX_u}}{\\mathcal Z_I(\\mathbb{Y})} \n\\nu^\\sharp_I(d\\mathbb{X} | \\mathbb{Y}) . \n\\end{equation}\nNote that this is a forward current on $\\Xi$ but by the above results it \ncan be obtained as the unique lift of its marginal on $\\mathcal{X}$ satisfying the\nidentification~(\\ref{eq:curr-constr-0}) between currents and stochastic\nintegrals. \n\n\\begin{remark}\n\\rm{\nThe specification~(\\ref{eq:spec2}) seems to depend only on the path\nand the currents appearing in the definition of the vector-field \n$w^C$ that describes the interaction with the boundary paths. The point \nof introducing measures and specifications on currents resides in the \nfact that we are not able to describe (\\ref{eq:spec2}) in terms of \npaths alone. The framework of stochastic currents is not the only\npossibility to solve this difficulty. A different way to proceed is \nconsidering directly rough paths and defining the measures and \nspecifications on the space of (step-2) rough paths, i.e., formally of\ncouples $(X,\\mathbb{X}^2)$, where $\\mathbb{X}^2$ is the twice iterated\nintegral associated with the paths $X$. This would solve the problem\nof stochastic integrals, which can then be defined as rough integrals, \nand with suitable growth conditions on the rough paths we would be \nalso allowed to define the interaction terms with boundary paths (over\nunbounded time intervals) and specifications similarly to that on the \ncurrents. Our approach is motivated essentially by the consideration \nthat currents are more basic objects than rough paths. We prefer to\nsee rough path theory as a tool for obtaining stochastic currents in\nuseful topologies. Indeed, in principle the construction of good \nversions of stochastic integrals can be carried out without recourse \nto rough paths~\\cite{FGGT,FGR}. \n}\n\\end{remark}\n\n\n\n\\section{Existence of Gibbs measures for Brownian currents}\n\\label{sect:cluster}\n\n\\subsection{Cluster representation}\n\\label{sec:cluster-rep}\n\nIn the following we will construct a Gibbs measure that is consistent\nwith the specification $\\{\\rho^\\sharp_I\\}_I$. This will be achieved \nby breaking up paths according to a sequence of bounded subintervals \nof the real line, and constructing Gibbs measures for bounded intervals. \nTaking limits over these Gibbs measures will result in a Gibbs measure \non $\\mathcal{X}$ whose lifted measure to $\\Xi$ is consistent with the given\nspecification. As mentioned before, a reasonably confining $V$ is \nneeded to make sure that the paths are not allowed to escape to \ninfinity with large probability. \n\nThe following notion of convergence will be used below to discuss\nGibbs measures. Let generally $E$ \nbe a metric space, and $C(\\mathbb{R},E)$ the space of continuous paths\n$\\{X_t; t \\in \\mathbb{R}\\}$ with values in $E$. For any bounded interval $I  \n\\subset \\mathbb{R}$ let $\\mathop{\\mathrm{{\\cal E}}}\\nolimits_I \\subset \\mathop{\\mathrm{{\\cal E}}}\\nolimits$ be a sub-$\\sigma$-field of the Borel \n$\\sigma$-field $\\mathop{\\mathrm{{\\cal E}}}\\nolimits$ of $E$ generated by the evaluations $\\{X_t: t \\in I\\}$. \nA sequence of probability measures $(m_n)_{\\{n \\in \\mathbb{N}\\}}$ on \n$C(\\mathbb{R},E)$ is said to converge locally weakly to the probability measure \n$m$ if for any such $I$ the restrictions $m_n|_{\\mathop{\\mathrm{{\\cal E}}}\\nolimits_I}$ converge weakly \nto the measure $m|_{\\mathop{\\mathrm{{\\cal E}}}\\nolimits_I}$. \n\nThe main result of this paper is the following\n\n\\begin{theorem}\nSuppose $V$ and $W$ satisfy the assumptions stated in Section \n\\ref{sec:potentials}. Take any unbounded increasing sequence \n$(T_n)_{n\\geq 0}$ of positive real numbers, and suppose $0 < |\\lambda| \n\\leq \\lambda^*$ with $\\lambda^*$ small enough. Then  the local weak \nlimit $\\lim_{n \\rightarrow \\infty} \\mu_{T_n} = \\mu$ exists on $\\mathcal{X}$ \nand does not depend on the choice of sequence $T_n$. Its unique lift \n$\\mu^\\sharp$ on $\\Xi$ is a Gibbs probability measure consistent with\nthe specification  $\\{\\rho^\\sharp_I\\}_I$.\n\\label{exist}\n\\end{theorem}\n\\begin{proof}\nWe develop a cluster expansion, i.e., choose the coupling parameter \n$\\lambda$ sufficiently small for being able to control the measure \nfor the interaction switched on ($\\lambda \\neq 0$) in terms of a \nconvergent perturbation series around the free case ($\\lambda = 0$).  \nThe theorem follows then through Propositions \\ref{p0}, \\ref{p2},\n\\ref{pclustest} and \\ref{prop:infinite-measure} below. \n\\end{proof}\n\n\\medskip\nTake a division of $[-T,T]$ into disjoint intervals $\\tau_k = (t_k, \nt_{k+1})$,  $k = 0,..., N-1$, with $t_0 = -T$ and $t_N = T$, each of \nlength $b$, i.e. fix $b = 2T\/N$; for convenience we choose $N$ to be \nan even number so that the origin is endpoint to some intervals. We \nbreak up a path $X$ into pieces $X_{\\tau_k}$ by restricting it to \n$\\tau_k$. The total energy contribution of the pair interaction then\nbecomes\n\\begin{equation}\nW_T(X) = \\frac{1}{2} \\sum_{i,j = 0}^{N-1} \n\\langle C^X_{\\tau_i},C^X_{\\tau_j}\\rangle_W  = \n\\sum_{0 \\leq i < j \\leq N-1} W_{\\tau_i,\\tau_j}\n\\label{sum}\n\\end{equation}\nwhere with the notation $\\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij} = \n\\langle C^X_{\\tau_i},C^X_{\\tau_j}\\rangle_W$ we have\n\\begin{equation}\nW_{\\tau_i,\\tau_j} \n= \\left\\{ \n\\begin{array}{ll}\n\\vspace{0.2cm} \n\\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij} + \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ji} & \\mbox{if $|i-j| \\geq 2$} \\nonumber \\\\\n\\vspace{0.2cm} \n\\frac{1}{2} (\\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ii} + \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{jj}) + \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij} + \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ji}  \n& \\mbox{if $|i-j| = 1$, and \n$i \\neq 0$, $j \\neq N - 1$} \\nonumber \\\\\n\\vspace{0.2cm} \n\\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij} + \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ji} + \\frac{1}{2} \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{00} \n& \\mbox{if $i = 0$ and $j = 1$} \\nonumber \\\\ \n\\vspace{0.2cm} \n\\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij} + \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ji} + \\frac{1}{2} \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{N-1 \\; N-1} \n& \\mbox{if $i = N-1$ and $j = N-2$.} \\nonumber \\\\\n\\end{array} \\right.\n\\end{equation}\nTo keep the notation simple we do not make explicit the $X$ \ndependence in these objects.\n\nBy using (\\ref{sum}) we obtain\n\\begin{equation}\ne^{-\\lambda W_T} = \\prod_{0 \\leq i < j \\leq N-1} \n(e^{-\\lambda W_{\\tau_i,\\tau_j}} + 1 - 1) =  1 + \\sum_{{\\cal R} \\neq \n\\emptyset} \\prod_{(\\tau_i,\\tau_j) \\in {\\cal R}} \n(e^{-\\lambda W_{\\tau_i,\\tau_j}} - 1).\n\\label{sum1}\n\\end{equation}\nHere the summation is performed over all nonempty sets of different \npairs of intervals, i.e. ${\\cal R} = \\{(\\tau_i,\\tau_j): \n(\\tau_i, \\tau_j) \\neq (\\tau_{i'},\\tau_{j'}) \\; \\mbox{whenever} \\; (i,j) \n\\neq (i',j') \\}$. \n\nA break-up of the paths involves a corresponding factorization of the \nreference measure into It\\^o bridges for each subinterval. Put $X_{t_k} = \nx_k$ for the positions at the time-points of the division, $\\forall \nk = 0,...,N$, with $-T = t_0 < t_1 < ... < t_N = T$. We write for a \nshorthand\n\\begin{equation}\n\\nu_T^{\\mathbf{x}}(\\cdot) \\equiv \\nu_T(\\, \\cdot \\,| X_{t_0} = x_0, \\ldots , \nX_{t_N} = x_N) = \\prod_{k=0}^{N-1} d\\nu_{\\tau_k}^{x_k,x_{k+1}}(\\cdot).\n\\label{mar}\n\\end{equation}\nLet $p_{t_0,...,t_N} (x_0,...,x_N)$ be the density with respect to \n$\\prod_{k=0}^N d\\omega(x_k)$, $d\\omega = \\Psi^2 dx$, of the joint \ndistribution of positions of the path $X$ recorded at the time-points \nof the division. \nBy Markovianness it follows that\n\\begin{eqnarray}\np_{t_0,...,t_N} (x_0,...,x_N) \n& = &\n\\prod_{k=0}^{N-1} \\pi_b(x_{k+1},x_k) = \\prod_{k=0}^{N-1} \n(\\pi_b(x_{k+1},x_k) - 1 + 1)  \n\\nonumber \\\\ \n& = &\n1 + \\sum_{\\cal S \\neq \\emptyset} \\prod_{k: \\tau_k \\in {\\cal S}} \n(\\pi_b(x_{k+1},x_k) - 1),\n\\nonumber\n\\end{eqnarray}\nwhere $\\pi_t$ is the transition kernel for the It\\^o diffusion given by\n(\\ref{pito}). The summation runs over all nonempty sets ${\\cal S} = \n\\{\\tau_k = (t_k,t_{k+1})\\}$ of different pairs of consecutive time-points. \n\nIn order to have a systematic control over these sums we introduce:\n\n\\begin{itemize}\n\\item[(1)] \n \\emph{Contours.} \\hspace{0.2cm} \nTwo distinct pairs of intervals $(\\tau_i,\\tau_j)$ and \n$(\\tau_{i'},\\tau_{j'})$ will be called directly connected and denoted \n$(\\tau_i,\\tau_j) \\sim (\\tau_{i'},\\tau_{j'})$ if one interval of the pair \n$(\\tau_i,\\tau_j)$ coincides with one interval of the pair $(\\tau_{i'},\n\\tau_{j'})$. A set of connected pairs of intervals is a collection \n$\\{(\\tau_{i_1},\\tau_{j_1}),..., (\\tau_{i_n},\\tau_{j_n})\\}$ in which \neach pair of intervals is connected to another through a sequence of \ndirectly connected pairs, i.e., for any $(\\tau_i,\\tau_j) \\neq \n(\\tau_{i'},\\tau_{j'})$ there exists $\\{(\\tau_{k_1},\\tau_{l_1}),..., \n(\\tau_{k_m},\\tau_{l_m})\\}$ such that $(\\tau_i,\\tau_j) \\sim \n(\\tau_{k_1},\\tau_{l_1}) \\sim ... \\sim (\\tau_{k_m},\\tau_{l_m}) \n\\sim (\\tau_{i'},\\tau_{j'})$. A maximal set of connected pairs of \nintervals is called a contour, denoted by $\\gamma$. We denote by \n$\\bar\\gamma$ the set of all intervals that are elements of the pairs \nof intervals belonging to contour $\\gamma$, and by $\\ga^*$ the set of \ntime-points of intervals appearing in $\\bar\\ga$. Two contours $\\ga_1, \n\\ga_2$ are disjoint if they have no intervals in common, i.e. \n$\\bar \\ga_1 \\cap \\bar \\ga_2 = \\emptyset$. Clearly, $\\cal R$ can \nbe decomposed into sets of pairwise disjoint contours: \n${\\cal R} = \\cup_{r \\geq 1} {\\cal R}_r$, where ${\\cal R}_r = \n\\{\\ga_1,...,\\ga_r\\}$ with $\\bar\\ga_i \\cap \\bar \\ga_j = \\emptyset$, \n$i \\neq j$; $i,j = 1,...,r$. \n\n\\item[(2)]  \\emph{Chains.} \\hspace{0.2cm} \nA collection of consecutive intervals $\\{\\tau_{j},\\tau_{j+1}...,\n\\tau_{j+k} \\}$, $j \\geq 0$, $j+k \\leq N-1$ is called a chain. \nAs in the case of contours, $\\bar \\rh$ and $\\rh^*$ mean the set of \nintervals belonging to the chain $\\rh$ and the set of time-points in \n$\\rh$, respectively. Two chains $\\rh_1, \\rh_2$ are called disjoint if \nthey have no common time-points, i.e. $\\rh_1^* \\cap \\rh^*_2 = \\emptyset$. \nDenote by $\\partial^-\\rh$ resp. $\\partial^+\\rh$ the leftmost resp. \nrightmost time-points belonging to $\\rh$. \n\n\\item[(3)]\n \\emph{Clusters.} \\hspace{0.2cm} \nTake a (non-ordered) set of disjoint contours and disjoint chains, \n$\\Ga = \\{\\gamma_1,...,\\gamma_r;\\varrho_1,...,\\varrho_s\\}$, with some \n$r \\geq 1$ and $s \\geq 0$. Note that such contours and chains may have \ncommon time-points. The notation $\\Ga^* = (\\cup_i \\ga^*_i) \\cup \n(\\cup_j \\rh_j^*)$ means the set of all time-points appearing as\nbeginnings or ends of intervals belonging to some contour or chain \nin $\\Gamma$. Also, we put $\\bar \\Ga = (\\cup_i \\bar\\ga_i) \\cup (\\cup_j \n\\bar\\rh_j)$ for the set of intervals appearing in $\\Ga$ through \nentering some contours or chains. $\\Gamma$ is called a cluster if \n$\\Ga^*$ is a connected collection of sets (in the usual sense), and \nfor every $\\varrho \\in \\Gamma$ we have that $\\partial^-\\rh, \n\\partial^+\\rh \\in \\cup_{j=1}^r\\ga^*_j$. This means that in a cluster \nchains have no loose ends. We denote by $\\mathop{\\mathrm{{\\cal K}}}\\nolimits_N$ the set of all clusters \nfor a given $N$. \n\n\\end{itemize}\n\nWith these notations the sum in (\\ref{sum1}) is then further expanded as\n\\begin{equation}\n\\sum_{{\\cal R} \\neq \\emptyset} \\prod_{(\\tau_i,\\tau_j) \\in \\cal R} \n(e^{-\\lambda W_{\\tau_i,\\tau_j}} - 1) = \\sum_{r \\geq 1} \n\\sum_{\\{\\gamma_1,...,\\gamma_r\\}} \\prod_{k = 1}^r \\prod_{(\\tau_i,\\tau_j) \n\\in \\gamma_k} (e^{-\\lambda W_{\\tau_i,\\tau_j}} - 1)\n\\label{sum2}\n\\end{equation}\nwhere now summation goes over collections $\\{\\gamma_1,...,\\gamma_r\\}$ \nof contours such that $\\bar\\ga_k \\cap \\bar\\ga_{k'} = \\emptyset$ \nunless $k = k'$. In a similar way (\\ref{mar}) appears in the form\n\\begin{equation}\n\\sum_{\\cal S \\neq \\emptyset} \\prod_{k: \\tau_k \\in {\\cal S}} \n\\left( \\pi_b(x_{k+1}, x_k) - 1 \\right) = \n\\sum_{s \\geq 1} \\sum_{\\{\\varrho_1,...,\\varrho_s\\}} \\prod_{j=1}^s \n\\prod_{k: \\tau_k \\in \\varrho_j} \\left(\\pi_b(x_{k+1},x_k) - 1 \\right).\n\\label{sum3}\n\\end{equation}\nHere $\\{\\varrho_1,...,\\varrho_s\\}$ is a collection of disjoint chains,\nand this formula justifies how we defined them. \n\nFor every cluster $\\Ga = \\{\\ga_1,...,\\ga_r;\\rh_1,...,\\rh_s\\} \\in \\mathop{\\mathrm{{\\cal K}}}\\nolimits_N$ \ndefine the function\n\\begin{equation}\n\\kappa_\\Ga = \\prod_{l=1}^r \\prod_{(\\tau_i,\\tau_j) \\in \\gamma_l} \n(e^{-\\lambda W_{\\tau_i,\\tau_j}} - 1) \\prod_{m=1}^s \n\\prod_{k: \\tau_k \\in \\rh_m}\\left(\\pi_b(x_{k+1},x_k) - 1\\right). \n\\label{kappa}\n\\end{equation}\nAlso, introduce the auxiliary probability measure \n\\begin{equation}\nd\\chi_N(X) = \\prod_{k=0}^{N-1} d\\nu_{\\tau_k}^{x_k,x_{k+1}}(X_{\\tau_k}) \n\\prod_{k=0}^N d\\omega(x_k),\n\\label{auxi}\n\\end{equation}\nand look at \n\\begin{equation}\nK_{\\Ga} = \\mathbb{E}_{\\chi} [\\kappa_\\Ga],\n\\label{aux}\n\\end{equation}\nwhere $\\chi$ is the unique extension over the real line of the family \nof consistent probabilities $\\{\\chi_N\\}_{N\\ge 1}$. Note that \n$\\int(\\pi_b(x_{k+1},x_k)-1) d\\omega(x_{k+1}) = \\int(\\pi_b(x_{k+1},x_k) \n-1) d\\omega(x_k) = 0$. This is the reason why from a cluster we rule \nout chains having loose ends; for any such chain $\\mathbb{E}_{\\chi_N} \n[\\kappa_\\Ga] = 0$.\n\nDefine \n\\begin{equation*}\n\\phi^T(\\Ga_1,...,\\Ga_n) = \\begin{cases}\n1 & \\text{if $n=1$}\\\\\n\\sum_{G \\in \\G^n} \\prod_{\\{i,j\\}\\in G} \n(- 1_{\\Ga^*_i \\cap \\Ga^*_j \\neq \\emptyset}) & \\text{if $n>1$},\n\\end{cases}\n\\end{equation*}\nwith $\\G^n$ the set of connected graphs on the vertex set\n$\\{1,\\dots,n\\}$. Note that $\\phi^T(\\Ga_1,...,\\Ga_n)=0$ if the graph \non the vertex set  $\\{1,\\dots,n\\}$ with edges $\\{i,j\\}$ drawn \nwhenever $\\Gamma^*_i \\cup \\Gamma^*_j \\neq \\emptyset$, is connected.\n\nBy putting (\\ref{sum2}), (\\ref{mar}), (\\ref{sum3}), (\\ref{kappa}) and \n(\\ref{aux}) together we obtain the cluster representation of the\npartition function $Z_T$.\n\\begin{proposition}\n\\label{p0}\nFor every $T = Nb\/2 > 0$ we have \n\\begin{equation}\n\\label{clexp}\nZ_T  = 1 + \\sum_{n \\geq 1} \\sum_{\\{\\Ga_1,...,\\Ga_n\\} \\in \\mathop{\\mathrm{{\\cal K}}}\\nolimits_N\n\\atop \\Ga^*_i \\cap \\Ga^*_j = \\emptyset, i \\neq j} \\prod_{l=1}^n \nK_{\\Ga_l}.\n\\end{equation}\nIf the activities $K_{\\Ga}$ satisfy the bound\n\\begin{equation}\n\\sum_{\\Ga \\in \\mathop{\\mathrm{{\\cal K}}}\\nolimits_N \\atop \\Ga^* \\ni 0, |\\bar\\Ga| = n} |K_{\\Ga}| \n\\;\\leq\\;  c \\; \\eta^n\n\\label{conv}\n\\end{equation}\nfor $\\eta > 0$ small enough, then the series above and at the right \nhand side of\n\\begin{equation}\n\\log Z_T = \\sum_{n \\geq 1} \\sum_{\\{\\Ga_1,...,\\Ga_n\\} \\in \\mathop{\\mathrm{{\\cal K}}}\\nolimits_N \n\\atop 0 \\in \\Ga_1^*} \\phi^T(\\Ga_1,...,\\Ga_n) \\prod_{l=1}^n K_{\\Ga_l}\n\\end{equation}\nare absolutely convergent, uniformly in $N$, and the latter one \ngives the logarithm of the partition function for the interval \n$[-T,T]$.\n\\end{proposition}\n\n\\noindent\nThe expression of the logarithm  and the absolute convergence of the \nsums are a general result of cluster expansion techniques, for details \nof proof see~\\cite{MM}.\n\n\n\\subsection{Convergence of cluster expansion}\n\n\\begin{proposition}\n\\label{p2}\nSuppose that there exist a function $D: \\mathbb{N}\\times\\mathbb{N} \n\\rightarrow (0,\\infty)$ and numbers $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits, C>0$ with\n$\n\\sup_{i\\in\\mathbb{N}} \\sum_{j\\in\\mathbb{N}} D(i,j) \\le C\n$\nsuch that for every $N > 0$ and every cluster \n$\\Ga = \\{\\ga_1,...,\\ga_r;\\rh_1,...,\\rh_s\\} \\in \\mathop{\\mathrm{{\\cal K}}}\\nolimits_N$, the bound \n\\begin{equation}\n|K_{\\Gamma}| \\; \\leq \\; \\left(\\prod_{l=1}^r \\prod_{(\\tau_i,\\tau_j) \n\\in \\gamma_l} \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i,j)\\right) \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits^{\\sum_{m=1}^s |\\bar\\rh_m|}\n\\label{clustest-abs}\n\\end{equation}\nholds. Then there is a constant $c > 0$ and a function \n$0 < \\eta(\\varepsilon) < 1$ with $\\eta \\to 0$ as $\\varepsilon \\to 0$ such that\n\\begin{equation}\n\\sum_{\\Ga \\in \\mathop{\\mathrm{{\\cal K}}}\\nolimits_N \\atop \\Ga^* \\ni 0, |\\bar\\Ga| = n} |K_{\\Ga}| \n\\;\\leq\\;  c \\; \\eta^n.\n\\label{convv}\n\\end{equation}\n\\end{proposition}\nThe function $D(i,j)$ will be specified in Proposition \n\\ref{pclustest} below. \n\n\\medskip\n\\begin{proof}\nWe put for a shorthand  \n$\\D(\\ga) = \\prod_{(\\tau_i,\\tau_j) \\in \\ga} \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i,j)$.\nConsider the function of complex variable $z$\n\\begin{equation}\nH(z;\\varepsilon) = \\sum_{\\Ga\\in\\mathop{\\mathrm{{\\cal K}}}\\nolimits_N: \\Ga^* \\ni 0} K_{\\Ga} z^{|\\bar \\Ga|} = \n\\sum_{\\Ga\\in\\mathop{\\mathrm{{\\cal K}}}\\nolimits_N: \\Ga^* \\ni 0 \\atop \\Ga \\supset\\; \\mbox{\\tiny{one contour}}} \nK_{\\Ga} z^{|\\bar\\Ga|} + \\sum_{\\Ga\\in\\mathop{\\mathrm{{\\cal K}}}\\nolimits_N: \\Ga^* \\ni 0 \\atop \\Ga \\supset\\;\n\\mbox{\\tiny{more than one contour}}}K_{\\Ga} z^{|\\bar\\Ga|}\n\\label{H}\n\\end{equation}\nWe show that for sufficiently small $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits > 0$ this is an analytic \nfunction of $z$ in a circle of radius $R(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)$ which diverges as \n$\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits \\to 0$. Moreover, we show that within this circle $H(z;\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)$ \nis uniformly bounded in $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits$. This will then imply (\\ref{convv}) by \nchoosing $\\eta(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits) = 1\/R(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)$. \n\nWe start by estimating the second sum; the first is simpler as it\ninvolves clusters having a single contour. Our strategy is first to\nbound it by a sum taken over graphs whose vertices are the contours of\nthe same cluster. The sums over graphs will then be bounded by sums\ntaken over trees.\n\n\\medskip\n\\noindent\n\\emph{\\textbf{Bounds by sums over graphs}} \\quad\nFor each $r \\geq 2$ consider in $\\mathop{\\mathrm{{\\cal K}}}\\nolimits_N$ those clusters $\\Ga$ that have $r$ \ncontours. For given $\\Ga \\in \\mathop{\\mathrm{{\\cal K}}}\\nolimits_N$ let $\\V_r = \\{\\ga_1,...,\\ga_r\\}$ be the \ncollection of these contours. We construct connected graphs $G$ by drawing \nedges between the elements of $\\V_r$ considered as vertex set. \nConnected graphs are those for which either $\\ga_i^* \\cap \\ga_j^* \\neq \\emptyset$ \nor there exists $\\rh_l \\in \\Ga$ such that $\\rh^*_l \\cap \\ga_i^* \\neq \\emptyset \n\\neq \\rh_l^* \\cap \\ga_j^*$. Let $\\G_r$ denote the set of all possible such graphs. \n\nConsider the collection of {\\it reduced} chains\n$\\{\\hat\\rh_1,...,\\hat\\rh_s\\}$ with the properties:\n\\begin{enumerate}\n\\item[(1)]\nfor every pair $\\{\\ga_i,\\ga_j\\} \\in G \\in \\G_r$, $\\ga_i^* \\cap \\ga_j^* = \\emptyset$, \nthere is at least one chain $\\hat\\rh_l$ of this collection connecting $\\ga_i$ and \n$\\ga_j$ (i.e. $\\overline{\\hat \\rh_l} \\cap \\bar \\ga_i = \\emptyset = \\overline{\\hat \n\\rh_l} \\cap \\bar \\ga_j$ and $\\hat \\rh_l^* \\cap \\ga_i^* \\neq \\emptyset \\neq \\hat \n\\rh_l^* \\cap \\ga_j^*$), and for any pair $\\{\\ga_i,\\ga_j\\} \\not \\in G$ such a chain \ndoes not occur;\n\\item[(2)]\n$\\cup_i \\bar\\ga_i \\cap \\cup_j \\overline{\\hat\\rh}_j = \\emptyset$  and \n$\\cup_j \\{\\partial^-{{\\hat\\rh}_j},\\partial^+ {{\\hat\\rh}_j}\\}  \\subset \n\\cup_i \\ga_i^*$; \n\\item[(3)]\n$0 \\in (\\cup_i \\ga_i^*) \\cup (\\cup_j {\\hat\\rh}_j^*)$;\n\\item[(4)]\neach chain $\\hat\\rh_k$ connects a pair $\\{\\ga_i,\\ga_j\\} \\in G \\in\n\\G_r$ \nor fills a gap within a contour $\\ga_i$.\n\\end{enumerate}\nWe call a collection of reduced chains compatible with graph $G$ if it\nsatisfies the conditions above and denote it \n$\\{\\hat\\rh_1,...,\\hat\\rh_s\\}_G$. Note that each $\\hat\\rh$ can join\nonly one pair of contours. A collection of reduced chains is then\nconstructed through the following steps:\n\\begin{enumerate}\n\\item[(1)]\nfirst remove all chains $\\rh_l \\in \\Ga$ for which $\\bar\\rh_l \\subset \\cup_{i=1}^r \n\\bar\\ga_i$;\n\\item[(2)]\nfor all remaining chains $\\rh_k \\in \\Ga$, $\\bar \\rh_k \\not\\subset \\cup_{i=1}^r \n\\bar\\ga_i$ remove all intervals from the set $\\bar\\rh_k \\cap (\\cup_{i=1}^r \\bar\\ga_i)$;\n\\item[(3)]\nof the remaining intervals form all possible collections of non-empty chains denoted \nby $\\{\\hat\\rh_1,...,\\hat\\rh_s\\}$.\n\\end{enumerate} \nThen by Proposition \\ref{clustest} we write\n\\begin{eqnarray}\n\\label{onecont}\n\\lefteqn{\n\\big|\\sum_{\\Ga: \\Ga^* \\ni 0 \\atop \\Ga \\supset\\;\n\\mbox{\\tiny{more than one contour}}} K_\\Ga z^{|\\bar\\Ga|}\\big| \n\\; \\leq \\label{est} } \\\\ &&\n\\sum_{r\\geq2} \\sum_{\\{\\ga_1,...,\\ga_r\\}} \\sum_{G \\in \\G_r} \n\\sum_{s \\geq 0} \\prod_{i=1}^r ((|z|(1+\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|))^{|\\bar\\ga_i|} \n\\D(\\ga_i) \\sum_{\\{\\hat\\rh_1,...,\\hat\\rh_s\\}_G \\atop \n\\mbox{\\tiny {$0 \\in \\cup_i \\gamma_i^* \\cup {\\hat\\rh}_i^*$}}} \n\\prod_{j=1}^s(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{|\\overline {\\hat\\rh_j}|}. \\nonumber\n\\end{eqnarray}\nNote that for fixed $\\{\\ga_1,...,\\ga_r\\}$ the collection\n$\\{\\hat\\rh_1,...,\\hat\\rh_m\\}$ can be obtained from many possible\ncollections of chains $\\{\\rh_1,...,\\rh_s\\}$. This gives the factor \n$(1 + \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{|\\bar\\ga_i|}$ appearing at the right hand side of \n(\\ref{est}). From now on we assume that $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|\\le 1$ so that we \ncan estimate this factor by $2^{|\\bar\\ga_i|}$.\n\n\nNow consider the last sum above involving the reduced chains.\nIn this sum, either $0$ belongs to a contour or some chains. In\nthe second case there is a factor of $(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,\\{\\gamma\\})}$\nappearing in the sum, so we can estimate the sum by\n\\begin{equation*}\n  \\begin{split}\n&  (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,{\\gamma})\/2} \\sum_{\\{\\hat\\rh_1,...,\\hat\\rh_s\\}_G \n\\atop \\mbox{\\tiny {$0 \\in \\cup_i \\gamma_i^* \\cup {\\hat\\rh}_i^*$}}}\n\\prod_{j=1}^s (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{|\\overline {\\hat\\rh_j}|\/2}\n \\le     \n \\sum_k (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,\\gamma_k)\/2}\n \\sum_{\\{\\hat\\rh_1,...,\\hat\\rh_s\\}_G \n\\atop \\mbox{\\tiny {$0 \\in \\cup_i \\gamma_i^* \\cup {\\hat\\rh}_i^*$}}}\n\\prod_{j=1}^s (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{|\\overline {\\hat\\rh_j}|\/2}.\n  \\end{split}\n\\end{equation*}\nFor each contour $\\gamma_i$ the sum over all reduced chains belonging\nto this contour cannot be larger than $2^{|\\overline{\\gamma_i}|}$\nsince the number of such chains is bounded by $\\overline{\\gamma_i}$\n(when every chain separates each two successive intervals in the\ncontour). Moreover for each couple of contours $(\\gamma_i,\\gamma_j)$\nthe contribution to the sum of the chains connecting them is given by\n$$\n2^{|\\overline{\\gamma_i}|+|\\overline{\\gamma_j}|} \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\gamma_i,\\gamma_j)\/2}\n$$\nsince there is at least one chain longer than\n$\\mathop{\\mathrm{dist}}\\nolimits(\\gamma_i,\\gamma_j)$ and the rest of the chains contribute\ninto the combinatorial prefactor. This gives\n\\begin{eqnarray}\n\\label{morecont-2}\n\\lefteqn{\n\\big|\\sum_{\\Ga: \\Ga^* \\ni 0 \\atop \\Ga \\supset\\;\n\\mbox{\\tiny{more than one contour}}}\nK_\\Ga z^{|\\bar\\Ga|}\\big| \\; \\leq \\label{estt} } \\\\ &&\n\\sum_{r\\geq2} \\sum_{\\{\\ga_1,...,\\ga_r\\}}\n \\sum_k  \\prod_{i=1}^r (8|z|)^{|\\bar\\ga_i|} \\D(\\ga_i) \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,\\gamma_k)\/2}  \\sum_{G \\in \\G_r}\n \\prod_{\\{\\gamma_i,\\gamma_j\\}\\in G} \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\gamma_i,\\gamma_j)\/2}.\n \\nonumber\n\\end{eqnarray}\n\n\\medskip\n\\noindent\n\\emph{\\textbf{Bounds by sums over trees}} \\quad\nWe use the tree-graph bound  (cf. Lemma 8, Ch. 2, Sect. 4 \nof \\cite{MM}) to get\n\\begin{equation}\n\\sum_{G\\in \\G_r} \\prod_{\\{\\ga_i,\\ga_j\\} \\in G} (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^\n{\\mathop{\\mathrm{dist}}\\nolimits (\\gamma_i,\\gamma_j)\/2} \n\\leq \n\\prod_{i,j =1}^r (1 + (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\gamma_i,\\gamma_j)\/2}) \n\\sum_{T \\in \\T_r} \\prod_{\\{\\ga_i,\\ga_j\\} \\in T} \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\gamma_i,\\gamma_j)\/2}\n\\label{mm8}\n\\end{equation}\nwhere $\\T_r$ is the set of trees on the vertex set\n$\\{\\gamma_1,\\dots,\\gamma_r\\}$. Moreover we have\n$$\n\\prod_{i,j =1}^r (1 + (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\gamma_i,\\gamma_j)\/2}) \n\\le\n2^{2 r} \\le 2^{2\\sum_i |\\overline\\gamma_i| },\n$$\nthus (\\ref{onecont}) is further estimated by\n\\begin{eqnarray}\n\\label{morecont-3}\n\\lefteqn{\n\\big|\\sum_{\\Ga: \\Ga^* \\ni 0 \\atop \\Ga \\supset\\;\n\\mbox{\\tiny{more than one contour}}}\nK_\\Ga z^{|\\bar\\Ga|}\\big| \\; \\leq \\label{esttt} } \\\\ &&\n\\sum_{r\\geq2} \\sum_{\\{\\ga_1,...,\\ga_r\\}}\n \\sum_k  \\prod_{i=1}^r (32|z|)^{|\\bar\\ga_i|} \\D(\\ga_i) \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,\\gamma_k)\/2}  \\sum_{T \\in \\T_r}\n \\prod_{\\{\\gamma_i,\\gamma_j\\}\\in G} (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits\n(\\gamma_i,\\gamma_j)\/2}.\n\\nonumber\n\\end{eqnarray}\n\n\nTake the trees over vertex set $\\{1,...,r\\}$ obtained through \n$\\ga_k \\mapsto k$, $\\forall k = 1,...,r$; denote them $\\tilde T$ \nand the set of all such trees by $\\tilde \\T_r$. Then we re-sum in\n(\\ref{morecont-3}):\n\\begin{equation}\n  \\begin{split}\n& \\mbox{r.h.s.}\\; (\\ref{morecont-3}) \\; \\leq\\\\\n& \\quad  \\sum_{r=2}^\\infty \\frac{1}{r!} \n\\sum_{\\tilde T \\in \\tilde \\T_r} \n\\sum_{i^*=1}^r \n\\sum_{(\\ga_1,...,\\ga_r)} \\prod_{i=1}^r (32|z|)^{|\\bar\\ga_i|} \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,\\ga_{i^*})\/2} \\D(\\ga_i) \n\\prod_{\\{i,j\\} \\in \\tilde T}\n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\ga_i,\\ga_j)\/2}.    \n  \\end{split}\n\\label{hae}\n\\end{equation}\nThe change of bracket indicates that the third sum here is performed \nover all ordered collections of disjoint contours. Fix an enumeration\nof $\\tilde T$ and pick its first element $i_0$. We estimate first \n\\begin{equation}\n\\sum_{(\\ga_1,...,\\ga_r)} \\prod_{i=1}^r (32|z|)^{|\\bar\\ga_i|} \\D(\\ga_i) \n\\prod_{\\{i,j\\} \\in \\tilde T} (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\ga_i,\\ga_j)\/2}.\n\\label{hva}\n\\end{equation}\nLet $j_0 \\neq i_0$ be an end vertex of tree $\\tilde T$ being joint \nonly with vertex $k_0$. Then\n\\begin{eqnarray}\n\\lefteqn{\n\\hspace{-1.5cm}\n\\sum_{\\ga_{j_0}} (32|z|)^{|\\bar\\ga_{j_0}|} \\D(\\ga_{j_0}) (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^\n{\\mathop{\\mathrm{dist}}\\nolimits(\\ga_{k_0},\\ga_{j_0})\/2}  \\label{i0}} \\\\ &&\n\\;\\leq\\;  \\sum_{\\tau'' \\in \\bar\\ga_{k_0}} \\sum_{\\ga_{j_0}} \n\\sum_{\\tau' \\in \\bar \\ga_{j_0}} (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\tau',\\tau'')\/2} \n(32|z|)^{|\\bar\\ga_{j_0}|} \\D(\\ga_{j_0}) \n\\nonumber  \\\\ &&\n\\;\\leq\\; \\sum_{\\tau'' \\in \\bar\\ga_{k_0}} \\sum_{\\tau'} \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\tau',\\tau'')\/2} \\sum_{\\ga_{j_0}: \\tau' \\in \\bar \n\\ga_{j_0}} (32|z|)^{|\\bar\\ga_{j_0}|} \\D(\\ga_{j_0}). \\nonumber \n\\end{eqnarray}\nHere we used that $(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\ga,\\ga')\/2} \n\\leq \n\\sum_{\\tau \\in \\ga, \\tau' \\in \\ga'} (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\tau,\\tau')\/2}$.\nBy using Lemma \\ref{lsumd} below and the bound\n\\begin{equation}\n\\sum_{k=2}^\\infty a^{k-1} k^m \\leq \\frac{2^{m}m! e a}{1 - e a}\n\\label{geo}\n\\end{equation}\nobtained via complex integration, we further estimate (\\ref{i0}) by\n\\begin{equation}\n\\label{j0}\n\\sum_{\\tau'' \\in \\bar\\ga_{k_0}} \\sum_{\\tau'} \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\tau',\\tau'')\/2} \n\\sum_{k=2}^\\infty (32|z|)^k (C\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)^{k-1} \n\\leq  \n\\frac{64 C |\\bar\\ga_{k_0}| \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|}{(1 - (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{1\/2})\n(1 - 32C \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)}. \n\\end{equation}\nFrom now on we choose $z$ such that $32C \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z| < 1$ holds.\n\nNext we go on by taking the next vertex of $\\tilde\\T$ in line, say\n$j_1 \\neq i_0$ connecting with $k_1$. We iterate the procedure for the\nnew tree obtained by deleting from $\\tilde T$ the vertex $j_0$ and\nedge $(j_0, k_0)$. If $j_1 \\neq k_0$, we get again an estimate of the\ntype (\\ref{j0}). If $j_1 = k_0$, we estimate\n\\begin{equation*}\n\\sum_{\\tau'' \\in \\ga_{k_1}} \\sum_{\\tau'} (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\tau',\\tau'')\/2} \n\\sum_{\\ga_{j_1}: \\tau' \\in \\ga}|\\bar\\ga_{j_1}|(32|z|)^{|\\bar\\ga_{j_1}|}\n\\D(\\ga)\n\\leq  \n\\const{q2} |\\bar\\ga_{k_1}| \\sum_{k=2}^\\infty (32|z|)^k \nk (C \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)^{k-1},\n\\end{equation*}\nwith some $\\defconst{q2} > 0$. Continuing this procedure inductively \nwe get after summation over $\\ga_{j_m}$, $j_m \\neq i_0$, connected \nto $\\ga_{k_m}$, the net contribution\n\\begin{eqnarray*}\n{\\const{q2}|\\bar\\ga_{k_m}|} \\sum_{k=1}^\\infty (32|z|)^k k^{l_{j_m} -1} \n(C\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)^{k-1} \n&\\leq&\n{\\const{q2}|\\ga_{k_m}|} \\sum_{k=2}^\\infty (\\const{q3}|z|)^k \nk^{l_{j_m}-1} (C\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)^{k-1} \\\\\n&\\leq&\n\\const{q9} \\; |\\ga_{k_m}||z| \\sum_{k=2}^\\infty ({\\const{q5}} \n\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|)^{k-1} k^{l_{j_1}-1}\n\\label{ind}\n\\end{eqnarray*}\nwhere $l_{j_m}$ is the degree of vertex $j_m$, i.e. the number \nof edges of $\\tilde T$ incident to $j_m$, and $\\defconst{q3},\n\\defconst{q4} > 0$, $\\defconst{q5} = C\\const{q3}$, $\\defconst{q9} \n= \\const{q3}\\const{q4}$ is the long sequence of constants. By using \n(\\ref{geo}) again, we estimate (\\ref{hva}) further for fixed \n$\\ga_{i_0}$ and $\\tilde T$ to get\n\\begin{equation*}\n\\sum_{\\ga_k: \\; k \\neq i_0 \\atop k = 1,...,r} \\prod_{\\{i,j\\} \\in \\tilde T} \n(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(\\tau_i,\\tau_j)\/2} \\prod_{i \\neq i_0} \\D(\\ga_i) (32|z|)^\n{|\\bar\\ga_i|} \\leq\n|\\bar\\ga_{i_0}| ^{l_{i_0}} \\prod_{k \\neq i_0} 2^{l_{j_k}} (l_{j_k}-1)! \\; \n(k_3 \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|^2)^{r-1}  \n\\end{equation*}\nwhere we used that $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z| \\leq 1\/(\\const{q5} e)$. Thus we need \nfurthermore (see (\\ref{hae}))\n\\begin{eqnarray*}\n\\lefteqn{\n\\sum_{\\ga_{i_0}} |\\bar\\ga_{i_0}|^{l_{i_0}} \\D(\\ga_{i_0})\n(16|z|(1+\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|))^{|\\ga_{i_0}|} (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,\\ga_{i_0})\/2} } \\\\ &&\n\\hspace{0.3cm} \\leq \n\\sum_{\\tau} (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,\\tau)\/2} \\sum_{\\ga_{i_0}: \\tau \\in \\ga_{i_0}} \n|\\bar\\ga_{i_0}|^{l_{i_0}} \\D(\\ga_{i_0}) (32|z|)^{|\\ga_{i_0}|}.\n\\end{eqnarray*}\nBy a repetition of the arguments above we get \n\\begin{eqnarray*}\n\\sum_{\\ga_{i_0}: \\tau \\in \\ga_{i_0}} |\\bar\\ga_{i_0}|^{l_{i_0}}\n\\D(\\ga_{i_0}) (32|z|)^{|\\ga_{i_0}|}\n& \\leq &\n\\sum_{k=2}^\\infty k^{l_{i_0}} (C\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)^{k-1} (32|z|)^k \\\\\n& \\leq &\n\\const{q6} 2^{l_{i_0}} l_{i_0}! \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|^2,\n\\end{eqnarray*}\nwith $\\defconst{q6} > 0$. Summation over $\\tau$ gives\n$\n\\sum_\\tau (\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits|z|)^{\\mathop{\\mathrm{dist}}\\nolimits(0,\\tau)\/2} \\leq \\; \\const{q7}, \n$\nwith some $\\defconst{q7} > 0$, hence we finally obtain for fixed \n$\\tilde T$ and $i_0$\n\\begin{eqnarray*}\n2^{l_{i_0}} l_{i_0}! \\prod_{j_k \\neq i_0} 2^{l_k} (l_{j_k}-1)! \n\\const{q7} (\\const{q6} \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|^2)^{r-1}\n&\\leq&\n\\const{q7} (2^2 \\const{q6} \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|^2)^{r-1} \\prod_{k=1}^r l_{j_k}! \n\\end{eqnarray*}\nwhere we used the fact $\\sum_{k=1}^r l_{j_k} = 2(r-1)$ for trees.  \nAn upper bound on the number of trees with vertices $\\{1,...,r\\}$ \nand degrees $\\{l_1,...,l_r\\}$ is \\cite{MM}\n\\begin{equation}\n\\frac{2^{r-2} (r-2)!}{\\prod_{j=1}^r l_j!}. \n\\label{inc}\n\\end{equation}\nMoreover, the number of collections $\\{l_1,...,l_r\\}$ such that $l_i > \n0$ and $\\sum_i l_i = 2(r-1)$ is bounded from above by $2^{2(r-1)}$.\nHence, by summing over $i_0$ and combining this estimate with\n(\\ref{inc}), we get\n\\begin{equation}\n\\sum_{\\Ga: \\Ga^* \\ni 0 \\atop \\Ga \\supset\\;\\mbox{\\tiny{more than one\n      contour}}} K_\\Ga |z|^{|\\bar\\Ga|} \\leq c \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|^2\n\\end{equation}\nwith some constant $c > 0$. This completes the estimate of the second\nterm in (\\ref{H}). The first term there can be handled in a similar\nway with substantial simplifications due to the fact that only one \ncontour occurs in the clusters. \n\nIt is seen then that by choosing $z$ such that $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits |z|^2 \\leq\n\\mbox{const}$, the sum $\\sum_\\Ga K_{\\Ga} z^{|\\bar \\Ga|}$ converges\nand is bounded. Hence $H(z)$ is an analytic function within a circle \nof radius $R(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)$ with $R(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits) \\to \\infty$ as $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits \\to 0$, and is \nbounded by a constant independent of $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits$. Thus \n\\begin{equation}\n\\sum_{\\Ga: \\Ga^* \\ni 0 \\atop |\\bar\\Ga| = n} |K_{\\Ga}| \\;\\leq\\; \n\\mbox{const} \\; R(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)^{-n} \\; := \\;  \\mbox{const} \\; \\eta(\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits) ^n,\n\\end{equation}\nwith suitable constants.\n\\end{proof}\n\nFinally we show the lemma referred to in the proof above. \n\\begin{lemma}\n\\label{lsumd}\nThere is a constant $C > 0$ such that for any interval $\\tau$ \nand integer $k \\geq 2$ \n\\begin{equation}\n\\sum_{\\ga: \\bar\\ga \\ni \\tau \\atop |\\bar\\ga|=k} \\D(\\ga) \\; \\leq\\; \n(C\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)^{k-1}.\n\\label{sumd}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\\begin{equation}\n\\sum_{\\ga: \\tau \\in \\bar\\ga \\atop |\\bar\\ga| = k} \\D(\\ga) = \n\\sum_{\\{\\tau_1,...,\\tau_k\\}} \\sum_{G\\in \\hat\\G}\n\\prod_{\\{\\tau_i,\\tau_j\\} \\in G} \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i,j)\n\\end{equation}\nHere $\\hat\\G$ denotes the set of connected graphs with vertices \n$\\tau_1,...,\\tau_k$. Note that for fixed $i_0$ we have \n$\\sum_{\\tau_j: j \\neq i_0}  \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i_0,j) \\le C \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits$ with some \n$C > 0$. Thus by using (\\ref{mm8}) we find\n\\begin{equation}\n\\sum_{G\\in\\hat\\G} \\prod_{\\{\\tau_i,\\tau_j\\} \\in G}  \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i,j)\n\\leq \ne^{k C \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits} \\sum_{T\\in \\hat\\T} \\prod_{\\{\\tau_i,\\tau_j\\} \\in T} \n\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i,j) \n\\label{102}\n\\end{equation}\nwhere $\\hat\\T$ are trees for the same vertex set as for $\\hat\\G$. \nNext order the  collection $\\{\\tau_1,...,\\tau_k\\}$ further to get \n\\begin{equation}\n\\sum_{\\{\\tau_1,...,\\tau_k\\}} \\sum_{T \\in \\hat T} \\prod_{\\{i,j\\} \n\\in \\tilde \\T} \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i,j)  =\n\\frac{1}{k!} \\sum_{(\\tau_1,...,\\tau_k)}\\sum_{\\tilde T \\in\\tilde\\T} \n\\prod_{\\{i,j\\} \\in \\tilde T}  \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i,j),\n\\label{101}\n\\end{equation}\nwith the same $\\tilde \\T$ as previously. We then obtain inductively\n\\begin{equation}\n\\sum_{(\\tau_1,...,\\tau_k)} \\prod_{(\\tau_i,\\tau_j) \\in \\T} \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits D(i,j)\n\\leq (C \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)^{k-1}.\n\\end{equation}\nSince the number of trees having $k$ vertices is $k^{k-2}$ \\cite{MM}, \nby using Stirling's formula, (\\ref{102}) and (\\ref{101}) we complete \nthe proof of the lemma.\n\\end{proof}\n\n\n\\subsection{Cluster estimates}\nHaving the abstract cluster expansion at hand, we turn now to\nestablishing the bounds~(\\ref{clustest-abs}) on the cluster\nactivities. \n\\begin{proposition}\nThere exists $\\delta > 1$ and a function\n$\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits(\\lambda)<\\infty$ with $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits(\\lambda) \\to 0$ as $\\lambda \\to 0$ and\n$b = b(\\lambda)\\ge 1$ such that for every $N > 0$ and every cluster\n$\\Ga = \\{\\ga_1,...,\\ga_r;\\rh_1,...,\\rh_s\\} \\in \\mathop{\\mathrm{{\\cal K}}}\\nolimits_N$, the bound \n\\begin{equation}\n|K_{\\Gamma}| \\; \\leq \\; \\prod_{l=1}^r \\prod_{(\\tau_i,\\tau_j) \\in \n\\gamma_l} \\frac{\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits}{(1+b|i-j-1|)^\\delta }\\prod_{m=1}^s  \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits^{|\\bar\\rh_m|}\n\\label{clustest}\n\\end{equation}\nholds.\n\\label{pclustest}\n\\end{proposition}\nThe first product (over the contours) above is our $D(i,j)$ in\nProposition \\ref{p2} above, and it is readily seen that it satisfies\nthe condition given there.\n\n\\medskip\n\\begin{proof}\nBy H\\\"older inequality\n\\begin{eqnarray}\n|K_{\\Ga}| \n& \\leq & \n\\prod_{l=1}^r \\prod_{(\\tau_i,\\tau_j) \\in \\gamma_l} \\left(\\int \\mathbb{E}_{\\nu_T^{\\mathbf{x}}}\n\\left[|e^{-\\lambda W_{\\tau_i,\\tau_j}} - 1|^{n_{ij}}\\right]\nd\\omega^{\\otimes N}(\\mathbf{x}) \\right)^{1\/n_{ij}} \n\\times \\\\\n&& \\hspace{0.3cm} \\times \n\\prod_{m=1}^s \\prod_{k: \\tau_k \\in \\rh_m}\\left(\\int|\\pi_b(x_{k+1},x_k) - 1|^\\beta \nd\\omega(x_k)d\\omega(x_{k+1})\\right)^{1\/\\beta} \\nonumber\n\\end{eqnarray}\nwith suitable exponents. We choose $\\beta = 4$, $n_{ij} = A |i-j+1|^{\\Delta}$, \nwith $\\Delta > 1$ to be specified below. Taken with correct \nmultiplicities, we pick $A$ such that\n$$\n\\frac{2}{\\beta} + \\sum_{j \\in \\mathbb{N} \\atop j \\geq i}\\frac{2}{n_{ij}} =\n\\frac{1}{2} + \\frac{2}{A}\\sum_{k=1}^\\infty \\frac{1}{k^\\Delta} \\leq 1.\n$$\nThe first part of estimate (\\ref{clustest}) follows by Lemma \\ref{holderw}, \nthe second by Lemma \\ref{holderpfi} below. By choosing $b= - \\log |\\lambda|\/\n(\\Lambda+C)$ with suitable $C>0$, we have\n$\n\\lambda e^{Cb} = e^{-\\Lambda b} = |\\lambda|^{\\Lambda\/(\\Lambda+C)} =: \\mathop{\\mathrm{{\\varepsilon}}}\\nolimits,\n$\nthus the estimate (\\ref{clustest}) is finally obtained.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:trans}\nFor large enough $b > 0$ there is a constant $C > 0$ such that\n\\begin{equation}\n|\\pi_b(x,y) - 1| \\,\\leq\\, Ce^{-\\Lambda b},\n\\end{equation}\nuniformly in $x,y \\in \\mathbb{R}^d$, where $\\Lambda > 0$ is the spectral gap of the \nSchr\\\"odinger operator $H = H_0 + V$. \n\\label{holderpfi}\n\\end{lemma}\n\\begin{proof}\nBy assumption the potential is chosen so that $H$ is intrinsically \nultracontractive, i.e., for each $b > 0$, $C_b = \\norm[L^{\\infty}(\\mathbb{R}^{2d})]{\\pi_b} \n< \\infty$. \n By the semigroup property of $\\pi_b$ and the fact that $\\int \\pi_b(x,y) d\\omega(y) = \n 1$ for each $x$, for $b > 2$ we have\n \\begin{equation}\n \\label{iucestimate}\n \\begin{split}\n |\\pi_b(x,y) - 1| \n & =  \n \\Big| \\int d\\xi \\int d\\eta \\,  \\pi_1(x,\\xi)\\Psi^2(\\xi) (\\pi_{b-2}(\\xi,\\eta) - 1) \n \\Psi^2(\\eta) \\pi_1(\\eta,y) \\Big| \\\\\n & \\leq  \n C_1^2 \\int d\\xi \\int d\\eta \\, \\Psi(\\xi) | \\hat\\pi_{b-2}(\\xi,\\eta) - \n \\Psi(\\xi)\\Psi(\\eta)| \\Psi(\\eta)  \\\\\n & \\leq  \n C_1^2 \\left( \\int d\\xi \\int d\\eta (\\hat\\pi_{b-2}(\\xi,\\eta) - \\Psi(\\xi) \\Psi(\\eta))^2 \n \\right)^{1\/2}  \\\\\n & = \n C_{1}^2 e^{-(b-2)(E_1-E)} \\left(\\sum_{k\\geq 2} e^{-2(b-2)(E_k- E_1)} \\right)^{1\/2} \n\\leq  \n Ce^{-\\Lambda b},\n  \\end{split}\n  \\end{equation}\nwhere $E = \\inf \\mathop{\\mathrm{Spec}}\\nolimits H$.\nThe last but first step comes about as follows. $\\norm[L^{\\infty}(\\mathbb{R}^{2d})]{\\pi_b} < \\infty$ \nimplies that $\\hat\\pi_b \\in L^2(\\mathbb{R}^{2d},dx)$ for each $b > 0$. Thus $e^{-bH}$ is a \nHilbert-Schmidt operator for each $b>0$, in particular $H$ has a purely discrete spectrum \nwith eigenvalues $E < E_{1} \\leq E_{2} \\leq \\ldots$. With $P_\\Psi$, the projection onto \nthe subspace of $L^2(\\mathbb{R}^d,dx)$ spanned by $\\Psi$, the last equality gives the Hilbert-Schmidt \nnorm of $e^{-(b-2)H} - P_\\Psi$.\n\\end{proof}\n\n\\medskip\n\\begin{lemma}\nAssume that $\\beta > 3$. Then there exists $\\delta > 1$ and\nconstants $\\defconst{ggg}, \\defconst{ggg2} < \\infty$ such that,  \nfor $\\lambda$ small enough\n\\begin{equation}\n\n\\begin{split}\n\\left(\\int \\mathbb{E}_{\\nu_T^{\\mathbf{x}}}\\left[ |e^{\\lambda\nW_{\\tau_i,\\tau_j}}-1|^{n_{ij}}\\right] d\\omega^{\\otimes N}\n(\\mathbf{x}) \\right)^{1\/n_{ij}} \n& \\le\n\\const{ggg}  \\lambda  e^{C b}  (1+b|i-j-1|)^{\\delta}\n\\end{split}\n\\end{equation}\nfor all $i,j$ and $b \\ge 1$.\n\\label{holderw}\n\\end{lemma}\n\\begin{proof}\nNote that the conditional expectation $\\mathbb{E}_{\\nu_T^{\\mathbf{x}}}\n[|e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{n_{ij}}]$ depends only on \n$x_i,x_{i+1},x_{j},x_{j+1}$. Thus we can write with a slight \nabuse of notation,\n$$\n\\int \\mathbb{E}_{\\nu_T^{\\mathbf{x}}} \n\\left[|e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{n_{ij}}\\right] \nd\\omega^{\\otimes N}(\\mathbf{x}) =\n\\int \\mathbb{E}_{\\nu_T^{\\mathbf{x}}} \n\\left[|e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{n_{ij}}\\right] \nd\\omega^{\\otimes 4}(x_i, x_{i+1}, x_j, x_{j+1})\n$$\nwhen $|i-j|>1$, otherwise one integral must be ignored in this \nexpression. Then by (\\ref{eq:bridge-2}) the problem reduces to \nestimates on the multiple Brownian bridge $\\widehat\\mathcal{W}_T^{\\mathbf{x}}$:\n\\begin{equation*}\n  \\begin{split}\n& A = \\int \\mathbb{E}_{\\nu_T^{\\mathbf{x}}} \n      \\left[|e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{n_{ij}}\\right] \n       d\\omega^{\\otimes 4}(x_i,x_{i+1},x_j,x_{j+1}) \\\\ \n& \\le\n\\int \\frac{\\mathbb{E}_{\\widehat\\mathcal{W}_T^{\\mathbf{x}}} \n      \\left[|e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{n_{ij}} \n       e^{-V_{\\tau_i}-V_{\\tau_j}}\\right]}\n       {Z_b \\Psi(x_i) \\Psi(x_{i+1}) \\pi_{b}(x_i,x_{i+1})}\n     \\frac{\\Pi_{b}(x_i,x_{i+1}) \\Pi_{b}(x_j,x_{j+1})}\n       {Z_b \\Psi(x_j) \\Psi(x_{j+1}) \\pi_{b}(x_j,x_{j+1})} \n       d\\omega^{\\otimes 4}(x_i,x_{i+1},x_j,x_{j+1}) \\\\ \n&  \\le \nC Z_b^{-2} \\int \\mathbb{E}_{\\widehat\\mathcal{W}_T^{\\mathbf{x}}} \n      \\left[|e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{n_{ij}} \n       e^{-V_{\\tau_i}-V_{\\tau_j}}\\right]\n       \\Pi_{b}(x_i,x_{i+1}) \\Pi_{b}(x_j,x_{j+1}) \\times \\\\ \n& \\qquad \\times \n\\Psi(x_i) \\Psi(x_{i+1}) \\Psi(x_j) \\Psi(x_{j+1})\ndx_i dx_{i+1} dx_{j} dx_{j+1},\n  \\end{split}\n\\end{equation*}\nwhere we used Lemma \\ref{holderpfi} and chose $b$ large enough \nso that\n$\n\\sup_{x,y} |\\pi_{b}(x,y)-1|\\le C e^{-\\Lambda b} \\le 1\/2\n$.\nFor $i+1<j$ notice that $\\Psi(x_{i+1})\\Psi(x_{j+1}) \\le C$. Then \nby integrating with respect to $x_{i+1},x_{j+1}$ we remove the \nconditional expectation and obtain\n\\begin{equation}\n\\label{eq:W-est-1}\n  \\begin{split}\nA\n& \\le C \n\\int \\widehat\\mathbb{E}_{x_i,x_j} \n \\left[|e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{n_{ij}} \n   e^{-V_{\\tau_i}-V_{\\tau_j}}\\right]\n   \\Psi(x_i) dx_i \\Psi(x_j) dx_j, \n \\end{split}\n\\end{equation}\nwhere now $\\widehat\\mathbb{E}_{x_i,x_j}$ denotes expectation over the two\npieces $\\mathbb{X}_{\\tau_i}$ and $\\mathbb{X}_{\\tau_j}$ weighted by $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x_i}_{\\tau_i} \n\\otimes \\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x_j}_{\\tau_j}$, i.e. two independent Wiener measures \nstarting at $x_i$ and $x_j$, respectively. \n\nNext we estimate the expectation in (\\ref{eq:W-est-1}),\n\\begin{equation*}\n  \\begin{split}\n&  \\widehat\\mathbb{E}_{x_i,x_j} \\left[|e^{\\lambda\n      W_{\\tau_i,\\tau_j}}-1|^{n} e^{-V_{\\tau_i}-V_{\\tau_j}}\\right]\n \\le      \\widehat\\mathbb{E}_{x_i,x_j} \\left[|\\lambda\n      W_{\\tau_i,\\tau_j}|^n e^{n |\\lambda\n      W_{\\tau_i,\\tau_j}|} e^{-V_{\\tau_i}-V_{\\tau_j}} \\right] \\\\ \n& \\qquad\n  \\le    |\\lambda|^n  \\left(\\widehat\\mathbb{E}_{x_i,x_j} \\left[|\n      W_{\\tau_i,\\tau_j}|^{2n}\\right]\\right)^{1\/2} \\left(\\widehat\\mathbb{E}_{x_i,x_j} \\left[ e^{4 n |\\lambda\n      W_{\\tau_i,\\tau_j}|}\\right]\\right)^{1\/4}  \\left(\\widehat\\mathbb{E}_{x_i,x_j} \\left[e^{-4 V_{\\tau_i}-4 V_{\\tau_j}} \\right]\\right)^{1\/4}.\n  \\end{split}\n\\end{equation*}\nSince $V$ is of Kato-class, we have the uniform bound\n$$\n \\left(\\widehat\\mathbb{E}_{x_i,x_j} \\left[e^{-4 V_{\\tau_i}-4 V_{\\tau_j}}\n \\right]\\right)^{1\/4} \\le \\sup_x  \\left(\\mathbb{E}_{x} \\left[e^{-4\n V_{[0,b]}} \\right]\\right)^{1\/2} \\le C e^{C b},\n$$\nwith some $C > 0$. Furthermore, write\n$\n W_{\\tau_i,\\tau_j} = \\int_{\\tau_i} dX_t \\int_{\\tau_j} dX_s g_{ts}\n$\n(see (\\ref{eq:J-rewriting}) below) and estimate the double integral below\nby using the Burkholder-Davis-Gundy inequality,\n\\begin{equation*}\n  \\begin{split}\n\\widehat\\mathbb{E}_{x_i,x_j} \\left[|\n      W_{\\tau_i,\\tau_j}|^{2n}\\right] & \\le c_{n} \\widehat\\mathbb{E}_{x_i,x_j} \\left[\\left|\n      \\int_{\\tau_i} dt\\left | \\int_{\\tau_j} dX_s g_{ts}\\right|^2\n      \\right|^{n}\\right]  \n\\\\  & \\le c_{n} |\\tau_i|^{n-1} \\int_{\\tau_i} dt \\widehat\\mathbb{E}_{x_i,x_j} \\left[\n     \\left | \\int_{\\tau_j} dX_s g_{ts}\\right|^{2n} \\right]      \n\\\\  & \\le c_{n}^2 |\\tau_i|^{n-1} |\\tau_j|^{n-1} \\int_{\\tau_i} dt  \\int_{\\tau_j} ds \\widehat\\mathbb{E}_{x_i,x_j} \\left[\n     |g_{ts}|^{2n} \\right]      \n\\\\  & \\le c_{n}^2 b^{2n} C^n (1+b|i-j-1|)^{-2 n \\beta}.\n  \\end{split}\n\\end{equation*}\nNow we estimate also the exponential of the energy. By $e^{|x|} \\le 2\n\\cosh x$ we get\n \\begin{equation*}\n  \\begin{split}\n \\widehat\\mathbb{E}_{x_i,x_j} \\left[ e^{4 n |\\lambda\n      W_{\\tau_i,\\tau_j}|} \\right] \\le  \\widehat\\mathbb{E}_{x_i,x_j} \\left[ e^{4 n \\lambda\n      W_{\\tau_i,\\tau_j}}  \\right]    +    \\widehat\\mathbb{E}_{x_i,x_j}\n      \\left[ e^{- 4 n \\lambda\n      W_{\\tau_i,\\tau_j}}  \\right].   \n  \\end{split}\n\\end{equation*}\nEach of the expectations in the right hand side can be similarly \nestimated by using Lemma~\\ref{lemma:final-exp-bound} below:\n\\begin{equation*}\n  \\begin{split}\n\\widehat\\mathbb{E}_{x_i,x_j} \\left[ e^{4 n |\\lambda\n      W_{\\tau_i,\\tau_j}|} \\right] \\le  (1-n^2 a^2)^{-\\const{b}} \\le C\n  \\end{split}\n\\end{equation*}\nwith \n$\na = \\const{a}\n    \\lambda b   (1+ b |i-j-1|)^{-\\beta} \n$ and $na \\le 1\/2$.\nHence, by making use these inequalities (uniform in $x_i,x_j$), \nand the fact that $\\Psi \\in L_1(\\mathbb{R}^d,dx)$, we arrive at\n\\begin{equation}\n \\label{eq:ewbound}\n\\begin{split}\n& \n\\left(\\int \\widehat\\mathbb{E}_{x_i,x_j}  \n\\left[ |e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{n_{ij}}\n e^{-V_{\\tau_i}-V_{\\tau_j}}\\right]  \n\\Psi(x_i) dx_i  \\Psi(x_j) dx_j \\right)^{1\/n_{ij}} \\\\\n&\n\\hspace{1cm} \\le \nC c_{n_{ij}}^{1\/n_{ij}}\\lambda be^{C b}(1+b|i-j-1|)^{-\\beta}. \n\\end{split}\n\\end{equation}\nBy using the estimate $c_k \\le (2k)^{2k}$ for the universal \nconstant in the Burkholder-Davis-Gundy inequality, we furthermore\nobtain\n$$\n c_{n_{ij}}^{1\/n_{ij}}  (1+b|i-j-1|)^{-\\beta} \\le (2n_{ij})^2\n (1+b|i-j-1|)^{-\\beta} .\n$$\nRecall that $n_{ij} = A |i-j|^\\Delta$ with $\\Delta >1$ and choose \n$\\Delta <\\beta$ so that $an_{ij} \\to 0 $ as $|i-j|\\to\\infty$ and \ncondition $an_{ij}\\le 1\/2$ is satisfied uniformly in $i,j$ for \n$\\lambda$ small enough. Thus there is a constant $\\defconst{fff} > \n0$ such that\n\\begin{equation}\n\\begin{split}\n\\text{r.h.s. (\\ref{eq:ewbound})}\n\\, \\le \\,\n\\const{fff}  \\lambda b^{2-2\\Delta} e^{C b}(1+b|i-j-1|)^{2\\Delta-\\beta}\n\\, \\le \\, \n\\const{fff}  \\lambda  e^{C b}(1+b|i-j-1|)^{-\\delta}, \n\\end{split}\n\\end{equation}\nfor all $|i-j| > 1$ and $b \\ge 1$, where $\\delta = \\beta-2\\Delta>1$\n(this is possible by choosing $\\Delta>1$ but small enough). For the \ncases $|i-j|=1$ we can follow a similar strategy to estimate\n$$\n\\left(\\mathbb{E}_{\\nu_T^{\\mathbf{x}}} \\left[|e^{\\lambda W_{\\tau_i,\\tau_j}}-1|^{2n_{ij}} \ne^{-V_{\\tau_i}-V_{\\tau_j}}\\right]\\right)^{1\/2n_{ij}} \n\\le \n\\defconst{tt2} e^{\\defconst{tt1} b},  \n$$ \nwhere the constants do not depend on either $b$ or $n_{ij}$.\n\\end{proof}\n\n\\subsection{Energy estimates}\nHere we estimate $\\widehat \\mathbb{E}_{x,y} [e^{\\lambda W_{\\tau_i,\\tau_j}}\n]$.\nBelow we will prove two lemmas that give the basic estimates by\nmaking use of the following result to control exponential\nintegrability of stochastic integrals. The first lemma will be \noften used for controlling exponential moments of stochastic integrals.\n\\begin{lemma}\n\\label{lemma:exp-bound}\nIf $X$ is Brownian motion and $f$ is an $\\mathop{\\mathrm{{\\cal F}}}\\nolimits$-adapted process, we \nhave the bound\n\\begin{equation}\n  \\label{eq:exp-bound}\n\\mathbb{E} [e^{\\int_0^b f_s dX_s}] \\le \n\\left\\{\\mathbb{E} [e^{2 \\int_0^b |f_s|^2 ds} ] \\right\\}^{1\/2}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe proof is a combination of Cauchy-Schwartz and Girsanov Theorems:\n  \\begin{equation*}\n    \\begin{split}\n \\mathbb{E} [e^{\\int_0^b f_s dX_s}] &= \n \\mathbb{E} [e^{\\int_0^b f_s dX_s -  \\int_0^b |f_s|^2 ds +\n  \\int_0^b |f_s|^2 ds} ]\n\\\\ & \\le  \\left\\{\\mathbb{E} [e^{2 \\int_0^b f_s dX_s - \\frac{1}{2} \\int_0^b\n |2 f_s|^2 ds}]\\right\\}^{1\/2}\\left\\{\\mathbb{E} [e^{\n 2 \\int_0^b |f_s|^2 ds} ]    \\right\\}^{1\/2}\n\\\\ & \\le \\left\\{\\mathbb{E} [e^{\n 2 \\int_0^b |f_s|^2 ds} ]    \\right\\}^{1\/2}.\n    \\end{split}\n  \\end{equation*}\n\\end{proof}\n\n\\medskip\n\\noindent\n\\emph{\\textbf Estimates for separated intervals} \\quad\nWe turn to estimating exponentials of energy contributions in \n(\\ref{sum}) by starting with pairs of intervals that are not \nadjacent. Let thus $i>j+1$; in this case the exponent has the \nform $W_{\\tau_i,\\tau_j} = \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij} + \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ji}$ with\n\\begin{equation}\n  \\label{eq:J-rewriting}\n\\mathop{\\mathrm{{\\cal J}}}\\nolimits_{i,j} =  \\langle C^X_{\\tau_i},C^X_{\\tau_j}\\rangle_W = \n\\int_{t_i}^{t_i+b} dX_t \\int_{t_j}^{t_j+b} d X_s\\, W(X_t-X_s,t-s).  \n\\end{equation}\nNote that  under the\nmeasure $\\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x_i}_{\\tau_i} \\otimes \\mathop{\\mathrm{{\\cal W}}}\\nolimits^{x_j}_{\\tau_j}$\nthe two currents  $\\mathbb{X}_{\\tau_i}$ and $\\mathbb{X}_{\\tau_j}$ are independent \nand the interaction energy can be written as a  double stochastic\nIt\\^o integral and estimated by using tools borrowed from stochastic \nanalysis.\n\\begin{lemma}\n\\label{lemma:final-exp-bound}\nLet $i>j+1$. There exist positive constants $\\defconst{a},\\defconst{b}$ \nsuch that whenever\n\\begin{equation}\n  \\label{eq:cond-bound-1}\na := \\lambda  \\const{a}  b (1+ |t_i-t_j-b|)^{-\\beta} < 1,  \n\\end{equation}\nwe have\n\\begin{equation*}\n  \\begin{split}\n\\widehat \\mathbb{E}_{x,y} [e^{\\lambda W_{\\tau_i,\\tau_j}}] &\n\\le \\left[1-a^2\\right]^{-\\const{b}}.\n  \\end{split}\n\\end{equation*}  \n\\end{lemma}\n\\begin{proof}\nWe have\n\\begin{equation*}\n  \\begin{split}\n\\left(\\widehat \\mathbb{E}_{x,y} [e^{\\lambda W_{\\tau_i,\\tau_j}}]\\right)^4 &\n\\le\n\\left(\\widehat \\mathbb{E}_{x,y} [e^{2\\lambda \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij}}]\\right)^2 \n\\left(\\widehat \\mathbb{E}_{x,y} [e^{2\\lambda \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ji}}]\\right)^2. \n  \\end{split}\n\\end{equation*}\nBy using (\\ref{eq:exp-bound}) we obtain\n\\begin{equation*}\n  \\begin{split}\n\\left(\\widehat \\mathbb{E}_{x,y} [e^{\\lambda \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij}}]\\right)^2 &\\le\n\\widehat \\mathbb{E}_{x,y} \\left[\\exp\\left(2 \\lambda^2 \\int_{t_i}^{t_i+b} dt\n  \\left|\\int_{t_j}^{t_j+b}  d X_s W(X_t-X_s,t-s)\n  \\right|^2\\right)\\right]\n\\\\ &\\le \\int_{t_i}^{t_i+b} \\frac{dt}{b}\n\\widehat \\mathbb{E}_{x,y} \\left [\\exp\\left(2 b \\lambda^2 \n  \\left|\\int_{t_j}^{t_j+b}  d X_s W(X_t-X_s,t-s) \\right|^2\\right)\\right ]\n\\\\ &\\le \\int_{t_i}^{t_i+b} \\frac{dt}{b}\n\\widehat \\mathbb{E}_{x,y} \\mathbb{E}_G \\left[\\exp\\left(2 \\lambda\n  G \\sqrt{b}\\int_{t_j}^{t_j+b}  d X_s W(X_t-X_s,t-s)\\right)\\right],\n  \\end{split}\n\\end{equation*}\nwith $G$ a normal random variable defined on a new probability space\nand $\\mathbb{E}_G$ the related Gaussian expectation. Then, using again \n(\\ref{eq:exp-bound}) yields\n\\begin{equation*}\n  \\begin{split}\n\\left(\\widehat \\mathbb{E}_{x,y} [e^{\\lambda W_{\\tau_i,\\tau_j}}]\\right)^2 &\n\\le \\int_{t_i}^{t_i+b} \\frac{dt}{b} \n\\left(\\widehat \\mathbb{E}_{x,y} \\mathbb{E}_G \\left[\\exp\\left(8 \\lambda^2 G^2 \nb \\int_{t_j}^{t_j+b} ds |W(X_t-X_s,t-s)|^2 \\right)\\right]\\right)^{1\/2}.\n  \\end{split}\n\\end{equation*}\nThe assumptions on $W$ give\n\\begin{equation*}\n  \\begin{split}\n  \\int_{t_j}^{t_j+b} ds |W(Y_t-Z_s,t-s)|^2 & \\le \n  \\int_{t_j}^{t_j+b} ds  \\frac{C}{(1+|t-s|)^{2\\beta}}\n \\le {\\const{c}} b (1+|t_i-t_j-b|)^{-2\\beta}\n  \\end{split}\n\\end{equation*}\nfor some constant $\\defconst{c}$,\nwhere we used that $t \\in [t_i,t_i+b]$. Hence\n\\begin{equation*}\n  \\begin{split}\n\\left(\\widehat \\mathbb{E}_{x,y} [e^{2\\lambda \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij}}]\\right)^2 &\n\\le  \\left(\n\\mathbb{E}_G \\left[\\exp\\left[32 \\lambda^2 \\const{c} b^2 \n(1+|t_i-t_j-b|)^{-2\\beta} G^2  \\right]\\right]\\right)^{1\/2}.\n  \\end{split}\n\\end{equation*}\nThe Gaussian integration can be performed explicitly, yielding\n\\begin{equation*}\n  \\begin{split}\n\\widehat \\mathbb{E}_{x,y} [e^{2\\lambda \\mathop{\\mathrm{{\\cal J}}}\\nolimits_{ij}}] &\n\\le   \n \\left(1-32 \\lambda^2 \\const{c} b^2 (1+|t_i-t_j-b|)^{-2\\beta}\n    \\right)^{-d\/8},\n  \\end{split}\n\\end{equation*}\n as soon as\n$\n1-32 \\lambda^2 \\const{c} b^2(1+ |t_i-t_j-b|)^{-2\\beta} > 0.\n$\nThus the claim follows.\n\\end{proof}\n\n\\medskip\n\\noindent\n\\emph{\\textbf Estimates for adjacent intervals} \\quad\nThe interaction energy estimates between adjacent intervals are \ngiven by\n\n\\begin{lemma}\nFor all $i = 0,...,N-2$ we have\n$\n\\widehat \\mathbb{E}_{x,y} [e^{\\lambda W_{\\tau_i,\\tau_{i+1}}}] \\le\n\\defconst{diag} e^{\\defconst{diag2} b } < \\infty\n$\nfor sufficiently small $\\lambda$.\n\\end{lemma}\n\\begin{proof}\nBy using that $W(x,t)$ is bounded and arguments similar to those \nof the previous lemma, the required exponential integrability of\n$W_{\\tau_i,\\tau_{i+1}}$ easily follows, at least for sufficiently\nsmall $\\lambda$.\n\\end{proof}\n\n\\subsection{Properties of the cluster expansion}\nWe finally show how the convergence of the cluster expansion of \n$Z_T$ and $\\log Z_T$ seen in Proposition \\ref{p0} imply existence \nof a limit Gibbs measure $\\mu$. \n\nFor any subset $A \\subset \\mathbb{R}$ let\n$$\nZ^A_T = 1 + \\sum_{n\\geq 1} \n\\sum_{{\\Ga_1,...,\\Ga_r \\in \\C_N \\atop \\Ga^*_i \\cap \\Ga^*_j \n= \\emptyset, \\; i \\neq j} \\atop A \\cap (\\cup_i\\Ga^*_i) \n= \\emptyset} \\prod_{i=1}^r K_{\\Ga_i}\n$$\nand write $Z_T^\\Gamma := Z_T^{\\overline \\Gamma \\cup \\Gamma^*}$.\nBy the cluster expansion we have\n$$\n\\log Z^\\Gamma_T = 1 + \\sum_{n\\geq 1} \n\\sum_{{\\Ga_1,...,\\Ga_r \\in \\C_N } \\atop \\Ga^* \\cap (\\cup_i\\Ga^*_i) \n= \\emptyset} \\phi^T(\\Ga_1,...,\\Ga_n)\\prod_{i=1}^r K_{\\Ga_i}.\n$$\nMoreover we can define the correlation functions for the clusters by\n\\begin{equation}\n  \\label{eq:cluster-corr}\nf^\\Gamma_T = \\frac{Z_T^\\Gamma}{Z_T} = \\exp\\left( - \\sum_{n\\geq 1} \n\\sum_{{\\Ga_1,...,\\Ga_r \\in \\C_N } \\atop \\Ga^* \\cap (\\cup_i\\Ga^*_i) \n\\neq \\emptyset} \\phi^T(\\Ga_1,...,\\Ga_n)\\prod_{i=1}^r K_{\\Ga_i} \\right) \n\\end{equation}\nand let $f^\\Gamma = \\lim_{T\\to\\infty} f^\\Gamma_T $ as the limit\nexists by the cluster estimates above and general arguments of \ncluster expansion~\\cite{MM}. Moreover we have the uniform estimate\n\\begin{equation}\n  \\label{eq:cluster-corr-estimate}\n|f^\\Gamma_T| \\le 2^{|\\overline \\Gamma|}  \n\\end{equation}\nfor $\\lambda$ small enough (the constant $2$ can actually be\nreplaced with any number larger than $1$, provided $\\lambda$ is\nchosen correspondingly small). Then existence of the infinite time \nlimit for the measures $\\{\\mu_T\\}_T$  follows  easily and we have\n\\begin{proposition}\n  \\label{prop:infinite-measure}\nThe local limit $\\mu = \\lim_{T\\to\\infty} \\mu_T$ exists and satisfies \nthe equality\n\\begin{equation}\n  \\label{eq:limit-eq-cluster}\n\\mathbb{E}_{\\mu} [F_S] =\n\\mathbb{E}_{\\chi} [F_S] f^{S} + \\sum_{n\\geq 1} \n\\sum_{{\\Ga_1,...,\\Ga_r \\in \\C \\atop \\Ga^*_i \\cap \\Ga^*_j \n= \\emptyset, \\; i \\neq j} \\atop i: S \\cap \\Ga^*_i \n\\neq \\emptyset}\n \\mathbb{E}_{\\chi}[F_S \\prod_{i=1}^r \\kappa^{\\Ga_i}] \nf^{\\cup \\overline \\Ga}\n\\end{equation}\nfor any bounded, $\\mathcal{F}_S$-measurable function $F_S$, where \n$S$ is a finite union of intervals of the partition considered in \nthe cluster expansion. Moreover, the measure $\\mu$ is invariant \nwith respect to time shift.\n\\end{proposition}\n\\begin{proof}\nWe have\n\\begin{equation}\n\\mathbb{E}_{\\mu}[F_S] = \\lim_{T \\rightarrow \\infty} \n\\frac{\\int e^{-\\lambda W_T(X)} F(X) d\\nu_T(X)}\n{\\int e^{-\\lambda W_T(X)} d\\nu_T(X)} = \n\\lim_{T \\rightarrow \\infty} \\frac{Z_T(F)}{Z_T}.\n\\end{equation}\nBy the cluster expansion we are led to \n\\begin{eqnarray}\nZ_T(F) = \n\\mathbb{E}_{\\chi} [F_S] Z_T^{S} + \\sum_{n\\geq 1} \n\\sum_{{\\Ga_1,...,\\Ga_r \\in \\C_N \\atop \\Ga^*_i \\cap \\Ga^*_j \n= \\emptyset, \\; i \\neq j} \\atop i: S \\cap \\Ga^*_i \n\\neq \\emptyset}\n \\mathbb{E}_{\\chi}[F_S \\prod_{i=1}^r \\kappa^{\\Ga_i}] \nZ_T^{\\cup \\overline \\Ga}.\n\\label{ce}\n\\end{eqnarray}\nIf $F_S$ is bounded, standard arguments show that the series \non the right hand side is absolutely convergent uniformly in \n$N$ and thus~(\\ref{eq:limit-eq-cluster}) follows. Given the \nuniqueness of the limiting measure, its invariance with respect \nto time shifts is a direct consequence of the invariance of the \npotentials and of the It\\^o-measure~(for more details \nsee~\\cite{LM}).\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:equiv-mu-nu}\nLet $F\\in\\mathcal{F}_{[0,b]}$ be a positive random variable. \nThen\n$\n\\mathbb{E}_{\\mu}[F] \\leq C (\\mathbb{E}_{\\nu}[F^2])^{1\/2}      \n$.  \n\\end{corollary}\n\\begin{proof}\nBy using Proposition ~\\ref{prop:infinite-measure} for $S=[0,b]$ \n(which for fixed $N$ is the interval $\\tau_{N\/2}$ of the \npartition) we have\n$$\n\\mathbb{E}_{\\mu} [F_S] = \\mathbb{E}_{\\chi} [F_S]  \nf^S + \\sum_{\\Ga_0: \\bar\\Ga_0 \\cap [0,b] \\neq \\emptyset} \n\\mathbb{E}_{\\chi} [F_S \\kappa^{\\Ga_0}] f^{\\Ga_0},   \n$$\nwhere in the second term the sum is over the only  cluster which\ncan overlap with $S$. We have $|f^S| \\le 2$, $|f^{\\Ga_0}| \\le \n2^{|\\Ga_0^*|}$. On the other hand, by using Lemma~\\ref{lemma:trans} \nand choosing $b$ large enough to ensure that \n$\\sup_{x,y}|\\pi_b (x,y)-1|\\le 1\/2$, we have\n\\begin{equation}\n\\mathbb{E}_{\\chi} [F_S] = \\int F_S(X) \\frac{d\\nu^{x_0,x_1}(X)}\n{\\pi_b(x_0,x_1)} \\pi_b (x_0,x_1) d\\omega(x_0) d\\omega(x_1)  \n\\; \\leq \\;  \nC \\; \\mathbb{E}_{\\nu} [F_S]. \n\\nonumber\n\\end{equation}\nFurthermore,\n$\n\\mathbb{E}_{\\chi^N} [\\kappa_a^{\\Ga_0}] \\leq \n\\left(\\mathbb{E}_{\\chi^N} [\\xi_a]\\right)^{1\/2} \\; \n\\left(\\mathbb{E}_{\\chi^N} [(\\kappa^{\\Ga_0})^2]\\right)^{1\/2}\n$\nand by the same arguments as in Proposition~\\ref{p2} above we \nobtain the bound\n$$\n\\sum_{\\Ga_0: [0,b]\\in \\bar \\Ga_0 } [ \\mathbb{E}_{\\chi} \n(\\kappa^{\\Ga_0})^2]^{1\/2} \\; 2^{|\\bar\\Ga_0|} \\; \\leq \\; \n\\mbox{const}\n$$\nwith some constant.  Hence we get that\n$\n\\mathbb{E}_{\\mu}[F_S] \\le C\n\\mathbb{E}_{\\nu}[F_S] + C\n(\\mathbb{E}_{\\nu}[F_S^2])^{1\/2}      \n$,\nwhich implies the claim.\n\\end{proof}\n\n\\begin{theorem}\n\\label{th:mu-lift}\nThere exists a unique forward current $\\mu^\\sharp$ on $\\Xi$ \nsuch that its $\\Xi$-marginal is $\\mu$. Moreover, under $\\mu$ \nwe have $\\mathcal{N}_\\alpha(X) < \\infty$ almost surely and \nthe boundary currents are well defined under $\\mu^\\sharp$.\n\\end{theorem}\n\\begin{proof}\nCorollary~\\ref{cor:equiv-mu-nu} implies that the measure $\\mu$ \nis absolutely continuous with respect to $\\nu$, thus the almost \nsure events of $\\nu$ carry over to $\\mu$ and we can consider \nthe lifted measure $\\mu^\\sharp$. This further implies that\n$$\n\\mathbb{E}_\\mu[N_{[k,k+1]}(X)^3] \n\\le \n\\left(\\mathbb{E}_\\nu[(N_{[k,k+1]}(X))^6]\\right)^{1\/2} \n\\le \nC, \n$$\nindependently of $k\\in\\Z$ and then $\\mathbb{E}_\\mu[\\mathcal{N}_\n\\alpha(X)] < \\infty$ for any $\\alpha > 1$. This last \ncondition guarantees the existence of the boundary currents \nunder the measure $\\mu^\\sharp$.\n\\end{proof}\n\n\\section{Properties of the Gibbs measure}\n\n\\subsection{Dependence on boundary conditions and DLR uniqueness}\n\nUniqueness in DLR sense means that for any increasing sequence of real \nnumbers $\\{T_n\\}_n$, $T_n \\uparrow \\infty$, and any corresponding \nsequence of boundary conditions $\\{Y_n\\}_n \\subset \\Xi$ we have \n$\\mathbb{E}_{\\rho^\\sharp_{T_n}(\\cdot |\\mathbb{Y}_n)}[F_B] \\to \\mathbb{E}_\\mu[F_B]$, \nfor every bounded $B \\subset \\mathbb{R}$, and each bounded and local (i.e., \nmeasurable with respect to $\\mathop{\\mathrm{{\\cal F}}}\\nolimits_B$) function $F_B$ on $\\Xi$. However,\nsuch a strong statement cannot be made in this context and we have to\nrestrict the class of allowed boundary conditions to be able to\ncontrol the limit. Fix $\\alpha > 1$ and let\n\\begin{equation}\n\\Xi_a = \\{Y \\in \\Xi : \\mathcal{N}_\\alpha(Y) \\le a \\},\n\\quad \\Xi_* = \\cup_{a > 0} \\Xi_a\n\\end{equation} \nbe the set of \\emph{allowed boundary conditions} carrying full \n$\\mu^\\sharp$ measure. Then we have\n\\begin{theorem}\n\\label{th:uniq}\nFor any $a>0$ the measure $\\mu$ is unique in DLR sense for any \nsequence of boundary conditions $(\\mathbb{Y}_n)_n$ in $\\Xi_a$, i.e.\n\\begin{equation}\n\\lim_{n\\to\\infty} \\mathbb{E}_{\\rho^\\sharp_{T_n}(\\cdot|\\mathbb{Y}_n)}[F_B] =\n\\mathbb{E}_\\mu[F_B].\n\\label{dlruniq}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nWe consider the class of bounded local functions $F_S$ on $\\Xi$ indexed \nby bounded intervals $S \\subset \\mathbb{R}$ (that is, $F_S$ is measurable with \nrespect to $\\mathop{\\mathrm{{\\cal F}}}\\nolimits_S$). It suffices to prove that for any increasing sequence \n$\\{T_n\\}$, $T_n \\to \\infty$, and any corresponding sequence of boundary \nconditions $(\\mathbb{Y}_n)_n \\subset \\Xi_a$ (\\ref{dlruniq}) holds for arbitrary \n$F_S$ of the above class. To show this we express the conditional \nexpectations appearing above in terms of the cluster representation. We \nsuppose without loss that $S$ consists of a finite union of intervals of \nthe partition of $[-T,T]$. \n\nFrom now on we follow the steps of the construction of the cluster\nrepresentation in Section \\ref{sec:cluster-rep}. Take the same partition \nof the interval $[-T,T]$ into disjoint segments as before. The interaction \nenergy can then be written as\n\\begin{equation}\nW_T(X|Y) = \n\\sum_{0 \\leq i < j \\leq N} W_{\\tau_i,\\tau_j}(X_{\\tau_i},X_{\\tau_j}) + \n\\sum_{0 \\leq i \\leq N-1} W_{\\tau_i,T}^Y (X_{\\tau_i}),\n\\end{equation}\nwith the same notations as before, and with \n\\begin{equation}\nW_{\\tau_i,T}^Y (X_{\\tau_i}) = 2 \\int_{\\tau_i}w^{C^{Y +}_{T}}(X_t,t) dX_t + \n2\\int_{\\tau_i}  w^{C^{Y-}_{-T}}(X_t,t) dX_t.   \n\\end{equation}\nBy (\\ref{a22}) the estimate \n\\begin{equation}\n\\label{a22y}\n\\sup_{x\\in \\mathbb{R}^d, t \\in \\tau} |w^{C^{Y \\pm}_{\\pm T}}(x,t)| \n\\leq \n\\frac{M_\\tau \\|C^{Y \\pm}_{\\pm T}\\|_{\\mathcal{D}'}}\n{\\left(\\mathop{\\mathrm{dist}}\\nolimits(\\tau,[-T,T]^c) + 1 \\right)^{\\beta-\\alpha}}\n\\end{equation}\neasily follows. (Here $[-T,T]^c = \\mathbb{R} \\backslash [-T,T]$ and \n$\\mathop{\\mathrm{dist}}\\nolimits(\\tau_k,[-T,T]^c) = \\min \\{kb, (N-k-1)b\\}$.) \n\nFix the positions $X_{t_0} = Y^-_{-T} = x_0$, $X_{t_1} = x_1$, ..., \n$X_{t_{N-1}} = x_{N-1}$, $X_{t_N} = Y^+_T = x_N$. Similarly to \n(\\ref{auxi}) introduce the auxiliary measure \n\\begin{equation}\nd\\chi_N^Y = \n\\prod_{k=0}^{N-1} \\frac{e^{-\\lambda W^Y_{\\tau_k,T}(X_{\\tau_k})}}\n{\\Z_{\\tau_k}^T(Y|x_k,x_{k+1})} d\\nu^{x_k,x_{k+1}}_b (X_{\\tau_k}) \n\\prod_{k=1}^{N-1} d\\omega(x_k) \n\\end{equation}\nwhere \n\\begin{equation}\n\\Z^T_{\\tau_k} (Y|x_k, x_{k+1}) = \\mathbb{E}_{\\nu^{x_k,x_{k+1}}_b} \n\\left[ e^{-\\lambda W_{\\tau_k,T}^Y (X_{\\tau_k})} \\right].\n\\end{equation}\nAlso, for every cluster $\\Ga$ consider the function $\\kappa_{\\Ga}^Y$ \ndefined similarly to (\\ref{kappa}). If $\\pm T \\not \\in \\Ga^*$, then \n$\\kappa^Y_\\Ga$ does not depend on $Y$. If $-T \\in \\Ga^*$ and\/or $T \\in \n\\Ga^*$, then $\\kappa^Y_\\Ga$ depends on $Y^-_{-T} = x_0$ and\/or $Y^+_T = \nx_N$, respectively. Next we define the weights\n\\begin{equation}\nK^Y_{\\Ga} = \\mathbb{E}_{\\chi^Y_N} [\\kappa^Y_\\Ga]\n\\end{equation}\nin the same manner as in (\\ref{aux}). The partition function $Z_T(Y)$ \ncan be expressed similarly to (\\ref{clexp}) with these altered objects. \nNote that\n\\begin{enumerate}\n\\item[(1)]\nfor sufficiently small $|\\lambda| \\neq 0$ the cluster estimate \n(\\ref{convv}) obtained in Proposition \\ref{p2} stays essentially \nvalid, i.e., \n\\begin{equation}\n\\label{eq:bound-KY}\n\\sum_{\\Ga: \\Ga^* \\ni 0 \\atop |\\bar \\Ga| = n} |K^Y_\\Ga| \\;\\leq\\; \nc \\eta'(\\lambda) ^n,\n\\end{equation}\nwith $\\eta'(\\lambda)$ going to zero as $\\lambda \\to 0$;\n\\item[(2)]\nfor any fixed $\\Ga$ we have\n$\n\\lim_{T\\to \\infty} \\kappa^Y_\\Ga = \\kappa_\\Ga\n$ and \n\\begin{equation}\n\\label{eq:convergence-KY}\n\\lim_{T\\to \\infty} K^Y_\\Ga = K_\\Ga,\n\\end{equation}\nboth uniformly convergent in $\\mathbb{Y} \\in \\Xi_a$. \n\\end{enumerate}\nThe proof of these statements goes by the same arguments  used \nin the previous section and it will be omitted. The \nbound~(\\ref{eq:bound-KY}) can be proven as in Lemma~\\ref{p2}.\nIndeed, by using Lemma~\\ref{lemma:exp-bound} and the bound~(\\ref{a22y}) \non the influence of the boundary current we have a handle to control \nthe exponential moments of  $W_{\\tau_i,T}^Y(X_{\\tau_i})$ in terms of \nthe norm of the boundary current and repeat the proof of \nLemma~\\ref{holderw} to obtain the necessary estimates on cluster \nactivities (with constants depending on $a$). A good control of the \nexponential moments is the key to obtain (\\ref{eq:convergence-KY}).\n\n\nThen in (\\ref{dlruniq}) we have\n\\begin{equation}\n\\mathbb{E}_{\\mu_T} [F_S|Y] = \\mathbb{E}_{\\chi^Y_T} \\left[F_S \\right] \n\\; f^S_T(Y) + \\sum_{n\\geq 1} \\sum_{\\{\\Ga_1,...,\\Ga_m\\}: \n\\Ga^*_i \\cap \\Ga_j^* \\neq \\emptyset \\atop \\Ga_i^* \\cap S \\neq 0, \n\\Ga_i \\subset [-T,T], i = 1,...,m} \n\\mathbb{E}_{\\chi^Y_T} \\left [F_S \\prod_{i=1}^m \\kappa^Y_T \\right] \\; \nf_T^{\\cup \\bar\\Ga}(Y)\n\\label{eee}\n\\end{equation}\nwith the same notations as in (\\ref{ce}) and $f^A_T(Y)= Z_T^A(Y)\/Z_T(Y)$.\n\nTake now a collection of intervals $\\{\\tau_i\\} = \\U$; the partition \nfunction $ Z_T^{\\U}(Y) := Z_{T}^{\\cup_{\\tau_i \\in \\U} \\tau^*_i} (Y)$ \ncan then be written like in~(\\ref{clexp}) except for changing $K_\\Ga$ \nfor $K^Y_\\Ga$.\n\\begin{lemma}\nFor sufficiently small $|\\lambda| \\neq 0$ we have the following \nproperties of $f^{\\U}_T(Y) := Z_T^\\U (Y)\/Z_T(Y)$. On the one hand,\n\\begin{equation}\n|f^{\\U}_T(Y)| \\;\\leq\\; 2^{|\\U|},\n\\label{zup}\n\\end{equation}\nwith $|\\U|$ denoting the number of intervals contained in $\\U$. \nOn the other hand,\n\\begin{equation}\n\\lim_{T \\to \\infty} f^{\\U}_T(Y) = f^{\\U}\n\\label{zap}\n\\end{equation}\nuniformly in $\\mathbb{Y} \\in \\Xi_a$. Moreover, $f^{\\U}$ also satisfies \n(\\ref{zup}) above. \n\\end{lemma}\n\\begin{proof}\nBoth statements are direct consequences of the \nbounds (\\ref{eq:bound-KY}) and of (\\ref{eq:convergence-KY})\ntogether with the cluster representation of the correlation \nfunctions~(\\ref{eq:cluster-corr}). By putting\n\\begin{equation}\n\\hat K_\\Ga^Y = \\left\\{ \n\\begin{array}{ll}\nK_\\Ga^Y & \\mbox{if $\\Ga$ lies inside $[-T,T]$} \\nonumber \\\\\n0 & \\mbox{otherwise} \n\\end{array} \\right.\n\\end{equation}\nand using dominated convergence we obtain that $f_{\\U}^Y \\to \nf_{\\U}$ uniformly as $T \\to \\infty$.\n\\end{proof}\n\nWe now return to the expression (\\ref{eee}). By ergodicity \nof the reference measure  \n\\begin{equation}\n\\lim_{T\\to \\infty} \\mathbb{E}_{\\chi_N^Y} [F_S] = \n\\mathbb{E}_{\\chi} [F_S],  \n\\end{equation}\nand hence the first term of (\\ref{eee}) converges to \n$\\mathbb{E}_{\\chi}[F_S] f_S$. By the same argument as above \nwe also obtain\n\\begin{equation}\n\\lim_{N \\to \\infty} \\mathbb{E}_{\\chi_N^Y} \n\\left[F_S \\prod_{i=1}^m \\kappa_{\\Ga_i}^Y \\right] \n= \n\\mathbb{E}_{\\chi}\\left[F_S \\prod_{i=1}^m \\kappa_{\\Ga_i}\\right] \n\\label{ele}\n\\end{equation}\nuniformly in $Y$, and \n\\begin{equation*}\n|\\mathbb{E}_{\\chi_N^Y} \n[F_S \\prod_{i=1}^m \\kappa_{\\Ga_i}^Y ]| \\;\\leq\\; (\\sup|F_S|) \n\\prod_{i=1}^m |K^Y_{\\Ga_i}|,\n\\label{maa}\n\\end{equation*}\nwhich implies\n\\begin{eqnarray*}\n\\sum_{\\{\\Ga_1,...,\\Ga_m\\} \\atop \\Ga^*_j \\cap S^* = \n\\emptyset, j = 1,...,m} \n\\left ( \\prod_{i=1}^m  |K^Y_{\\Ga_i}|\\right)  \n2^{|\\cup_i \\bar\\Ga_i|} 2^{|S^*|}\n& \\leq &\n2^{|S^*|} \\sum_{m=1}^{|S^*|} {|S^*| \\choose m} \n\\left( \\sum_{\\Ga: \\Ga \\ni 0} 2^{|\\bar\\Ga|} |K^Y_\\Ga| \\right)^m \\\\\n& \\leq &\n2^{|S^*|} \\sum_{m=1}^{|S^*|} {|S^*| \\choose m} \n\\left( \\sum_{n=2}^\\infty 2^n c \\eta(\\lambda)^n \\right)^m < \\infty.\n\\end{eqnarray*}\nHere $S^*$ denotes the time points occurring in $S$. By using now \nthis estimate together with (\\ref{ele}) and applying Lebesgue's \ndominated convergence theorem once again, we arrive at\n\\begin{equation}\n\\mathbb{E}_{\\chi} [F_S] f_S + \\sum_{m=1}^\\infty \n\\sum_{\\{\\Ga_1,...,\\Ga_m\\} \\atop \\Ga_i^* \\cap S^* \\neq\n\\emptyset, i = 1,...,m} \\mathbb{E}_{\\chi} \n\\left[F_S \\prod_{i=1}^m \\kappa_{\\Ga_i} \\right] \nf_{S \\cup (\\cup_i \\bar\\Ga_i)} = \\mathbb{E}_\\mu [F_S].\n\\end{equation}\n\\end{proof}\n\n\\begin{corollary}\nThe measure $\\mu^\\sharp$ satisfies the DLR equations\n$$\n\\int_\\Xi \\mathbb{E}_{\\rho^\\sharp_T(\\cdot|\\mathbb{Y})}[F_S] \nd\\mu^\\sharp(\\mathbb{Y}) = \\mathbb{E}_{\\mu^\\sharp}[F_S]\n$$  \nfor any $S<T$.\n\\end{corollary}\n\\begin{proof}\nNote that $|\\mathbb{E}_{\\rho^\\sharp_T(\\cdot|\\mathbb{Y})}[F_S]| \\le \\sup |F_S|$ \nand that $\\cup_{a>0} \\Xi_a$ has full $\\mu^\\sharp$-measure so that \nthe left hand side of the equality makes sense. Fix $a>0$. By \nTheorem~\\ref{th:uniq} we have\n$$\n\\int_\\Xi \\mathbb{E}_{\\rho^\\sharp_T(\\cdot|\\mathbb{Y})}[F_S] d\\mu^\\sharp(\\mathbb{Y}) \n= \n\\lim_{T_2\\to+\\infty}\\int_\\Xi \\mathbb{E}_{\\rho^\\sharp_T(\\cdot|\\mathbb{Y})}[F_S] \nd\\rho^\\sharp_{T_2}(\\mathbb{Y}|\\mathbb{Z})\n$$\nuniformly in $Z\\in \\Xi_a$. Then the DLR consistency of the \nspecification implies\n$$\n\\int_\\Xi \\mathbb{E}_{\\rho^\\sharp_T(\\cdot|\\mathbb{Y})}[F_S|\\mathbb{Y}] d\\mu^\\sharp(\\mathbb{Y}) = \n\\lim_{T_2\\to+\\infty} \\mathbb{E}_{\\rho^\\sharp_{T_2}(\\cdot|\\mathbb{Z})}[F_S] \n= E_{\\mu^\\sharp}[F_S],\n$$\nproving the statement.\n\\end{proof}\n\n\n\\subsection{Typical path configurations}\nIn this section we show that most of $\\mu$'s weight is\nconcentrated on paths that can be characterized by a growth \ncondition. The proof of this depends on a lemma which was\nalready shown in \\cite{LM}, however, we include it here for \nmaking the presentation more self-contained.   \n\\begin{theorem}\nSuppose (\\ref{iuccond}) holds with $a = b = s$, and\n$\\mu$ is a probability measure obtained by Theorem \\ref{exist} \nfor $V$ and $W$. Then there is a number $C > 0$ and a functional \n$R(X)$, such that \n\\begin{equation}\n|X_t| \\leq \\left(C \\log (|t| + 1)\\right)^{1\/(s+1)} + R(X)\n\\end{equation}\n$\\mu$-almost surely.\n\\label{tpb}\n\\end{theorem}\n\\begin{proof}\nThe strategy of proving this theorem is to derive the typical \nbehaviour of $\\mu$ from the typical behaviour of the reference \nprocess. This follows through Lemma \\ref{ldphi} below. Then\ncombining this lemma with Corollary~\\ref{cor:equiv-mu-nu} gives\n\\begin{equation*}\n  \\begin{split}\n\\mu\\left(\\max_{0 \\leq t \\leq 1} |X_t| \\geq a\\right)\n& \\;\\le \\; \nC \\left[\\mathbb{E}_\\nu(1_{\\max_{0 \\leq t \\leq 1} |X_t| \\geq a})\\right]^{1\/2}      \n\\le  c'  e^{-\\theta' a^{s+1}},\n  \\end{split}\n\\end{equation*}\nwith some constants $c',\\theta' > 0$. Thus under the stationary \nmeasure $\\mu$ \n\\begin{equation}\n\\mu\\left(\\max_{n \\leq t \\leq n+1} |X_t| \\; \\geq \\; \n(k\\log n)^{1\/(s+1)}\\right) \\; \\leq \\; \\mbox{const} \n\\frac{1}{n^{k \\theta'}}\n\\end{equation}\nholds. Choosing $k$ so that $k \\theta' > 1$, the \nBorel-Cantelli Lemma implies that $\\mu$-almost surely\n\\begin{equation}\n|X(t)| \\; \\leq \\; (k \\log t)^{1\/(s+1)}\n\\end{equation}\nfor $t \\geq T^*$, with $T^* = T^*(X)$ sufficiently large. \nWriting $R(X) = \\max_{|t| \\leq T^*} |X(t)|$ completes the proof. \n\\end{proof}\n\nFinally we prove the lemma used above. \n\\begin{lemma}\nLet $\\nu$ be the measure of the It\\^o-process for $V$ satisfying \n(\\ref{iuccond}) with exponent $s > 2$, and $a > 0$. Then \nthere exist $C > 0$ and $\\theta > 0$ such that\n\\begin{equation}\n\\nu\\left(\\max_{0 \\leq t \\leq 1} |X_t| \\geq a \\right) \n\\; \\leq C \\; e^{-\\theta a^{s+1}}.\n\\end{equation}\n\\label{ldphi}\n\\end{lemma}\n\\begin{proof}\nFor the underlying It\\^o-process we have the Dirichlet operator \non $L^2(\\mathbb{R}^d,d\\nu)$\n\\begin{equation}\nLf = -\\Delta f + 2 (\\nabla \\log \\Psi, \\nabla f)\n\\end{equation}\nand Dirichlet form \n\\begin{equation}\n{\\cal E}(f,f) = - \\int f \\Delta f d\\omega + \n2 \\int f (\\nabla \\log \\Psi, \\nabla f) d\\omega,\n\\end{equation}\nwith $d\\omega = \\Psi^2 dx$, as before. By using Varadhan's Lemma \n(see Lemma 1.12, \\cite{KV86}), for any $f \\in L^2(d\\omega)$ and \nevery $N > 0$  \n\\begin{equation}\n\\nu \\left(\\max_{0\\leq t\\leq 1} |f(X_t)| \\geq N\\right) \n\\; \\leq \\; \\frac{3}{N} \\sqrt{{\\cal E}(f,f) + (f,f)}\n\\label{kv}\n\\end{equation}\nholds. Choose $f = f_a := 1_{\\{x \\in \\mathbb{R}^d: |x| \\geq a\\}} * \\phi$ \nby picking a mollifier $\\phi$ (with $||\\phi||_\\infty < \\infty$) \nso that the above convolution is in the domain of $L$. This can be \nchosen so that the smoothing of the edges of the indicator function \ntakes place in a sphere $S(a)$ of radius $a$ centred at the origin, \ni.e., with a suitable $\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits > 0$ we take $f_a(x) = 1$ for $x \\in \\mathbb{R}^d \n\\backslash S(a+\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)$, $f_a(x) = 0$ for $x \\in S(a-\\mathop{\\mathrm{{\\varepsilon}}}\\nolimits)$, and $f_a$ \nis a sufficiently smooth function ${\\tilde f}_a$ otherwise. Denote \nthese three domains by $D_1$, $D_2$ and $D_3$, respectively. Setting \n$N=1$ in (\\ref{kv}) yields\n$$\n\\nu \\left(\\max_{0 \\leq t \\leq 1} |X_t| \\geq a \\right) \\leq \\; \n3 \\sqrt{||f_a||^2_{L^2(d\\omega)} + (f_a, L f_a)_{L^2(d\\omega)}}.\n$$\nMoreover, we have\n$$\n||f_a||^2_{L^2(d\\omega)} = \\int f_a^2 (x) d\\omega(x) = \n\\int_{D_1} d\\omega(x) + \\int_{D_3} {\\tilde f}_a^2(x) d\\omega(x).\n$$\nUnder the hypothesis on $V$ the standard estimate \n$\n\\Psi(x) \\le C e^{-\\theta |x|^{s+1}}\n$\nholds by Carmona's results \\cite{Ca78} for the ground state $\\Psi$, \nwith some $C,\\theta > 0$. This bound further leads to\n\\begin{equation}\n\\int_{D_1}d\\omega(x) \\; \\leq \\; c e^{-\\theta a^{s+1}},\n\\end{equation}\nwhere $c,\\theta > 0$ are independent of $a$. A similarly estimate is \nvalid for $D_3$. On the other hand, since ${\\tilde f}_a$ is smooth \nenough and $\\max_{D_3} \\{ |\\nabla {\\tilde f}_a|,|\\Delta {\\tilde f}_a|, \n\\Delta {\\tilde f}_a^2 \\} \\leq m < \\infty$, we get\n$$\n({\\tilde f}_a, L{\\tilde f}_a) \\leq c' e^{-\\theta a^{s+1}},\n$$\nwith suitable $c'> 0$. A similar estimate is obtained also for the \nremaining two domains. \n\\end{proof}\n\n\n\n\\subsection{Mixing properties}\nSince $\\mu$ is constructed in a way that offers no immediate \naccess to computations with this measure, it is important to \nderive further basic information on $\\mu$ by using the cluster\nexpansion. We give here one last result of this paper. \n\\begin{theorem}\nLet $F,G$ be two bounded functions, the first measurable with respect \nto $\\mathop{\\mathrm{{\\cal F}}}\\nolimits_I$, the second with respect to $\\mathop{\\mathrm{{\\cal F}}}\\nolimits_J$, where $I,J$ are distinct \nintervals of the partition considered in the cluster expansion above. \nThen the estimate on the covariance \n\\begin{eqnarray*}\n&& \\mathop{\\mathrm{cov}}\\nolimits_\\mu (F;G) = \\mathbb{E}_\\mu[F G] - \\mathbb{E}_\\mu[F] \\; \n\\mathbb{E}_\\mu[G] \\\\\n&& |\\mathop{\\mathrm{cov}}\\nolimits_\\mu \\; (F;G)| \\;\\leq \\; \\mbox{\\rm{const}} \\; \n\\frac{\\sup{|F|} \\sup{|G|}} {|t-s|^\\vartheta + 1}\n\\end{eqnarray*}\nholds, where $\\vartheta > 0$ and the constant prefactor is \nindependent of $F,G$.\n\\end{theorem}\n\n\\medskip\n\\begin{proof}\nFirst recall formula~(\\ref{eq:limit-eq-cluster}) which applied to $F$ \n(and similarly to $G$) gives\n\\begin{equation}\n\\mathbb{E}_\\mu [F] = \\mathbb{E}_\\nu [F] f^{I} + \n\\sum_{\\Ga_0: I \\cap \\Ga_0^* \\neq \\emptyset} K^{\\Ga_0}(F) f^{\\Ga_0^*},\n\\label{F}\n\\end{equation}\nwhere we let $K_{\\Ga_0}(F) = \\mathbb{E}_{\\chi} [F \\kappa_{\\Ga_0}] $. \nFurthermore, consider $f^A$ estimated as before like $|f^A| \\leq \n2^{|A|}$. For $A_1 \\cap A_2 = \\emptyset$ we have\n\\begin{equation}\n|f^{A_1 \\cup A_2} - f^{A_1} f^{A_2}| \\; \\leq \\; \\mbox{const} \\; \n\\frac{2^{|A_1| + |A_2|}}{\\mathop{\\mathrm{dist}}\\nolimits(A_1,A_2)^\\zeta},\n\\label{cost}\n\\end{equation}\nwith some $\\zeta > 0$. This estimate can easily be obtained by the \ngeneral results in \\cite{MM}. Now we  write\n\\begin{equation}\n\\begin{split}\n\\mathbb{E}_\\mu [F G] \n& =  \n\\mathbb{E}_\\nu [F] \\; \\mathbb{E}_\\nu [G] f^{I \\cup J} \\\\\n& \\qquad + \n\\sum_{\\Ga_1 \\atop  \\Ga^*_1 \\cap I \\neq \\emptyset , \\Ga^*_1 \\cap J \n= \\emptyset} \\mathbb{E}_{\\chi} [F \\kappa_{\\Ga_1}] \\; \n\\mathbb{E}_\\chi [G]  f^{J \\cup \\Ga^*_1}\n+ \\sum_{\\Ga_2 \\atop  \\Ga^*_2 \\cap J \\neq \\emptyset , \\Ga^*_2 \\cap I \n= \\emptyset} \\mathbb{E}_{\\chi} [G \\kappa_{\\Ga_2}] \\; \n\\mathbb{E}_\\chi [F]  f^{I \\cup \\Ga^*_2}\n\\\\\n& \\qquad + \n\\sum_{\\Ga_1,\\Ga_2 : \\Ga^*_1 \\cap \\Ga^*_2 = \\emptyset \\atop  \\Ga^*_2 \n\\cap J \\neq \\emptyset , \\Ga^*_2 \\cap I \\neq \\emptyset}  \n\\mathbb{E}_\\chi [F \\kappa_{\\Ga_1}] \\; \n\\mathbb{E}_{\\chi} [G \\kappa_{\\Ga_2}]  f^{\\Ga^*_1 \\cup \\Ga^*_2}\n\\sum_{\\Ga \\atop  \\Ga^* \\cap J \\neq \\emptyset , \\Ga^* \\cap I \\neq \n\\emptyset}  \\mathbb{E}_\\chi [F G \\kappa_{\\Ga}] f^{\\Ga^*}.\n\\end{split}\n\\end{equation}\nFrom here and (\\ref{F}) we obtain\n\\begin{equation}\n\\begin{split}\n\\mathop{\\mathrm{cov}}\\nolimits_\\mu(F; G) \n& = \n\\mathbb{E}_\\chi [F] \\; \\mathbb{E}_\\chi [G](f^{I\\cup J}-f^{I}f^{J}) \n\\\\ & \\qquad + \n\\sum_{\\Ga_1 \\atop  \\Ga^*_1 \\cap I \\neq \\emptyset , \\Ga^*_1 \\cap J \n= \\emptyset} \\mathbb{E}_{\\chi} [F \\kappa_{\\Ga_1}] \\; \n\\mathbb{E}_\\chi [G]  (f^{J \\cup \\Ga^*_1}- f^{J} f^{\\Ga^*_1})\n \\\\ & \\qquad \n+ \\sum_{\\Ga_2 \\atop  \\Ga^*_2 \\cap J \\neq \\emptyset , \\Ga^*_2 \\cap I \n= \\emptyset} \\mathbb{E}_{\\chi} [G \\kappa_{\\Ga_2}] \\; \n\\mathbb{E}_\\chi [F]  (f^{I \\cup \\Ga^*_2}-f^{I}f^{\\Ga^*_2})\n\\\\\n& \\qquad + \n\\sum_{\\Ga_1,\\Ga_2 : \\Ga^*_1 \\cap \\Ga^*_2 = \\emptyset \\atop  \\Ga^*_2 \n\\cap J \\neq \\emptyset , \\Ga^*_2 \\cap I \\neq \\emptyset}  \n\\mathbb{E}_\\chi [F \\kappa_{\\Ga_1}] \\; \\mathbb{E}_{\\chi} \n[G \\kappa_{\\Ga_2}]  (f^{\\Ga^*_1 \\cup \\Ga^*_2}-f^{\\Ga^*_1}f^{\\Ga^*_2})\n \\\\ & \\qquad\n+ \n\\sum_{\\Ga \\atop  \\Ga^* \\cap J \\neq \\emptyset , \\Ga^* \\cap I \\neq \n\\emptyset}  \\mathbb{E}_\\chi [F G \\kappa_{\\Ga}] f^{\\Ga^*}\n \\\\ &\\qquad\n- \n\\sum_{\\Ga \\atop  \\Ga^* \\cap J \\neq \\emptyset , \\Ga^* \\cap I \\neq \n\\emptyset}  \\mathbb{E}_\\chi [F ] \\mathbb{E}_\\chi [ G \\kappa_{\\Ga}] \nf^{\\Ga^*} f^I f^{\\Ga^*}\n \\\\ & \\qquad\n- \n\\sum_{\\Ga \\atop  \\Ga^* \\cap J \\neq \\emptyset , \\Ga^* \\cap I \\neq \n\\emptyset}  \\mathbb{E}_\\chi [F \\kappa_{\\Ga}] \\mathbb{E}_\\chi [ G ] \nf^{\\Ga^*} f^J f^{\\Ga^*}\n \\\\ & \\qquad \n- \\sum_{\\Ga_1,\\Ga_2 : \\Ga^*_1 \\cap \\Ga^*_2 \\neq \\emptyset \\atop  \n\\Ga^*_2 \\cap J \\neq \\emptyset , \\Ga^*_2 \\cap I \\neq \\emptyset}  \n\\mathbb{E}_\\chi [F \\kappa_{\\Ga_1}] \\; \\mathbb{E}_{\\chi} \n[G \\kappa_{\\Ga_2}]  f^{\\Ga^*_1}f^{\\Ga^*_2}\n\\end{split}\n\\end{equation}\nFor estimating the first four terms at the right hand side above we \nuse (\\ref{cost}) along with the bound\n\\begin{equation}\n|\\mathbb{E}_{\\mathop{\\mathrm{{\\cal P}}}\\nolimits} [F \\kappa_{\\Ga}]| \\; \\leq \\; \\frac{\\sup |F|}\n{(\\mathop{\\mathrm{diam}}\\nolimits \\Ga^*)^{\\zeta'} + 1} E_{\\Ga}(\\delta',\\varepsilon')\n\\end{equation}\n$i = 1, 2$, where $E_{\\Ga}(\\delta',\\varepsilon')$ is the function \nappearing at the right hand side of estimate~(\\ref{clustest}) with \nslightly modified entries ($\\delta',\\varepsilon'$ instead of $\\delta,\\varepsilon$; \n$\\delta' > 1$) so that (\\ref{convv}) still holds. Here $\\zeta' = \n\\delta - \\delta' > 0$, and we used in addition that \n\\begin{equation}\n\\frac{1}{(\\mathop{\\mathrm{diam}}\\nolimits \\Ga^*)^{\\zeta'} + 1} \\; \n\\frac{1}{\\mathop{\\mathrm{dist}}\\nolimits(I,\\Ga^*)^\\zeta + 1} \\; \\leq \\; \n\\frac{1}{\\mathop{\\mathrm{dist}}\\nolimits(I,J)^\\vartheta + 1}\n\\end{equation}\nwhenever $J \\cap \\Ga^* \\neq \\emptyset$ (similarly for $I$), and\n\\begin{equation}\n\\frac{1}{(\\mathop{\\mathrm{diam}}\\nolimits \\Ga^*_1)^{\\zeta'} + 1} \\; \n\\frac{1}{(\\mathop{\\mathrm{diam}}\\nolimits \\Ga^*_2)^{\\zeta'} + 1} \\; \n\\frac{1}{\\mathop{\\mathrm{dist}}\\nolimits(\\Ga^*_1,\\Ga^*_2)^\\zeta} \\; \\leq \\; \n\\frac{1}{\\mathop{\\mathrm{dist}}\\nolimits(I,J)^\\vartheta + 1}\n\\end{equation}\nfor $I \\cap \\Ga^*_1 \\neq \\emptyset$, $J \\cap \\Ga^*_2 \\neq \\emptyset$, \nand $\\vartheta = \\min \\{\\zeta,\\zeta'\\} > 0$.\n\nNext, in the fifth term above we use that $\\mathop{\\mathrm{diam}}\\nolimits \\Ga^* \\geq \\mathop{\\mathrm{dist}}\\nolimits(I,J)$ \nwhenever $I \\cap \\Ga^* \\neq \\emptyset$, $J \\cap \\Ga^* \\neq \\emptyset$, \nand that\n\\begin{equation}\n|\\mathbb{E}_{\\chi} [F G \\kappa_\\Ga]| \\; \\leq \\; \\sup|F| \\sup|G| \n\\frac{E_\\Ga(\\delta',\\varepsilon')} {(\\mathop{\\mathrm{diam}}\\nolimits \\Ga^*)^{\\zeta'} + 1}.\n\\end{equation}\nFor the remaining three terms in the sum above we apply the same argument. \nThus for the full sum the corresponding bounds become\n\\begin{equation}\n\\mbox{const} \\; \\frac{\\sup|F| \\sup|G|}{\\mathop{\\mathrm{dist}}\\nolimits(I,J)^\\vartheta + 1} \nE_{\\Ga_1}(\\delta',\\varepsilon') \nE_{\\Ga_2}(\\delta',\\varepsilon') \\; 2^{|\\Ga_1^*| + |\\Ga^*_2|}, \n\\end{equation}\nwhenever $I \\cap \\Ga^*_1 \\neq \\emptyset$, $J \\cap \\Ga^*_2 \\neq \\emptyset$, \nrespectively \n\\begin{equation}\n\\mbox{const} \\; \\frac{\\sup|F| \\sup|G|}{\\mathop{\\mathrm{dist}}\\nolimits(I,J)^\\vartheta + 1} \nE_{\\Ga}(\\delta',\\varepsilon') \\; 2^{|\\Ga^*|} \n\\end{equation}\nin the other cases. Then using  Proposition~\\ref{p2} we can prove \nboundedness of the sums over  $\\Ga_1, \\Ga_2$ or $\\Ga$, concluding the \nproof.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\@startsection{subsection}{3{\\@startsection{subsection}{3}\n\t\\z@{.5\\linespacing\\@plus.7\\linespacing}{.5\\linespacing}\n\t{\\bfseries\\itshape}} \\makeatother \n\\renewcommand{\\thesubsection}{\\thesection\\Alph{subsection}}\n\n\n\\makeatletter \\renewenvironment{proof}[1][\\proofname]{\n\t\\par\\pushQED{\\qed}\\normalfont\n\t\\topsep6\\mathfrak{p}@\\@plus6\\mathfrak{p}@\\relax\n\t\\trivlist\\item[\\hskip\\labelsep\\bfseries#1\\@addpunct{.}]\n\t\\ignorespaces}{\n\t\\popQED\\endtrivlist\\@endpefalse} \\makeatother\n\n\\def\\:{\\colon}\n\\numberwithin{equation}{section}\n\n\\renewcommand{\\bar}{\\overline}\n\n\n\\begin{document}\n\n\\title[Factorial affine surfaces with $\\kappa=0$ and trivial units]{Classification of smooth factorial affine surfaces \\\\ of Kodaira dimension zero with trivial units}\n\\subjclass[2010]{Primary: 14R05; Secondary: 14J26, 57R65, 57M99}\n\\keywords{affine surface, $\\mathbb{C}^{*}$-fibration, log Minimal Model program, knot surgery, Kirby diagram}\n\\author{Tomasz Pe\u0142ka}\n\\address{Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain}\n\\email{tpelka@bcamath.org}\n\\author{Pawe\u0142 Ra\u017any}\n\\address{Instytut Matematyki, Uniwersytet Jagiello\u0144ski, ul. \u0141ojasiewicza 6, 30-348 Krak\u00f3w, Poland}\n\\email{pawel.razny@uj.edu.pl}\n\\begin{abstract}\n\tWe give a corrected statement of \\cite[Theorem 2]{GM_k<2}, which classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. Denote the class of such surfaces by $\\mathcal{S}_{0}$. An infinite series of surfaces in $\\mathcal{S}_{0}$, not listed in \\cite{GM_k<2}, was recently obtained by Freudenburg, Kojima and Nagamine  \\cite{FKN_k0} as affine modifications of the plane. We complete their list to a series containing arbitrarily high-dimensional families of pairwise non-isomorphic surfaces in $\\mathcal{S}_{0}$. Moreover, we classify them up to a diffeomorphism, showing that each occurs as an interior of a $4$-manifold whose boundary is an exceptional surgery on a $2$-bridge knot \\cite{BW_surgery}. In particular, we show that $\\mathcal{S}_{0}$ contains countably many pairwise non-homeomorphic surfaces.\n\\end{abstract}\n\n\\maketitle\n\nWe work with complex algebraic varieties.\n\\section{Introduction}\n\nThe structure of smooth affine surfaces of non-general type is essentially understood, see \\cite[Chapter III]{Miyan-OpenSurf}. It can be summarized in a following way, which parallels the classical Enriques-Kodaira classification of smooth projective surfaces. Let $S$ be a smooth affine surface and let $\\kappa\\in \\{-\\infty,0,1,2\\}$ be its Kodaira--Iitaka dimension. If $\\kappa=-\\infty$ then $S$ admits a $\\mathbb{C}^{1}$-fibration \\cite[III.1.3.2]{Miyan-OpenSurf}. If $\\kappa=1$ then $S$ has a (canonical) $\\mathbb{C}^{*}$-fibration \\cite[III.1.7.1]{Miyan-OpenSurf}. The surfaces with $\\kappa=0$ are peculiar, but understandable: they are obtained in a controlled way from some specific minimal models, see \\cite[II.6.4]{Miyan-OpenSurf}.  \n\nLet $\\mathcal{S}_{\\kappa}$ be the class of smooth affine surfaces $S$ of Kodaira--Iitaka dimension $\\kappa$, such that $\\mathbb{C}[S]^{*}=\\mathbb{C}^{*}$ and $\\mathbb{C}[S]$ is factorial, i.e.\\ all line bundles on $S$ are trivial. It is known \\cite[III.2.2.1]{Miyan-OpenSurf} that $S\\in \\mathcal{S}_{-\\infty}$ if and only if $S\\cong \\mathbb{C}^{2}$. The above description of affine surfaces with $\\kappa<2$ suggests that it is the class $\\mathcal{S}_{0}$ where one could look for unusual examples.\n\nIn \\cite[Theorem 2]{GM_k<2}, Gurjar and Miyanishi claimed that, up to an isomorphism, there are at most two surfaces in $\\mathcal{S}_{0}$. This result, however, was recently proved wrong by Freudenburg, Kojima and Nagamine \\cite{FKN_k0}, who constructed an infinite series of pairwise non-isomorphic surfaces in $\\mathcal{S}_{0}$. \n\nThe aim of this article is to complete the description of $\\mathcal{S}_{0}$ by proving the following theorem.\n\n\\begin{thm}\\label{THM}\nLet $\\mathcal{S}_{0}$ be the class of smooth affine surfaces $S$ such that the ring $\\mathbb{C}[S]$ is factorial, $\\mathbb{C}[S]^{*}=\\mathbb{C}^{*}$ and the logarithmic Kodaira--Iitaka dimension of $S$ is zero. Then $S\\in \\mathcal{S}_{0}$ if and only if\n\\begin{equation*}\nS\\cong S_{p_1,p_2}:= \\operatorname{Spec} \\mathbb{C}[x_1,x_2][(x_2x_1^{-\\deg p_1}-p_1(x_1^{-1}))x_1^{-1},(x_1x_2^{-\\deg p_2}-p_2(x_2^{-1}))x_2^{-1}]\n\\end{equation*}\nfor some monic polynomials $p_1,p_2\\in \\mathbb{C}[t]$. Moreover, $S_{p_1,p_2}\\not \\cong S_{p_1',p_2'}$ for $\\{p_1,p_2\\}\\neq \\{p_1',p_2'\\}$.\n\\end{thm}\n\n\\begin{rem}\\label{rem:literature}\n\t\\begin{enumerate}\n\t\t\\item\\label{item:GM} The two surfaces constructed in \\cite[Theorem 2]{GM_k<2} are both isomorphic to $S_{1,1}$, see \\cite[Theorem 4.4]{FKN_k0}.\n\t\tThis surface was extensively studied in \\cite[\\S 8]{Kal-Ku1_AL-theory}.\n\t\t\\item\\label{item:FKN} For $n\\geq 1$, the surface $V_{n}=\\operatorname{Spec} \\mathbb{C}[x,y][(x-1)\/(x^{n}y-1)]$ constructed in \\cite{FKN_k0} is isomorphic to $S_{t^{n-1},1}$ via $\\mathbb{C}[x_1,x_2][(x_2-1)x_1^{-n},(x_1-1)x_2^{-1}]\\ni (x_1,x_2)\\mapsto (x,1-x^{n}y)\\in \\mathbb{C}[x,y][(x-1)\/(x^{n}y-1)]$.\n\t\\end{enumerate}\n\\end{rem}\n\n\\begin{figure}[htbp]\n\t\\begin{tabular}{ccc}\n\t\t\\begin{subfigure}[t]{0.35\\textwidth}\n\t\t\t\\includegraphics[scale=0.25]{figures\/S_kl}\n\t\t\t\\caption{$p_1,p_2\\neq 1$}\n\t\t\t\\label{fig:Skl}\n\t\t\\end{subfigure}\n\t\t&\n\t\t\\begin{subfigure}[t]{0.35\\textwidth}\n\t\t\t\\includegraphics[scale=0.25]{figures\/S_k0}\n\t\t\t\\caption{$p_1\\neq 1$, $p_2=1$}\n\t\t\t\\label{fig:Sk0}\n\t\t\\end{subfigure}\n\t\t&\n\t\t\\begin{subfigure}[t]{0.15\\textwidth}\n\t\t\t\\includegraphics[scale=0.25]{figures\/S_00}\n\t\t\t\\caption{$p_1=p_2=1$}\n\t\t\t\\label{fig:S00}\n\t\t\\end{subfigure}\n\t\\end{tabular}\n\t\\caption{Standard boundaries of the surfaces $S_{p_1,p_2}$ from Theorem \\ref{THM}.}\n\t\\label{fig:boundaries}\n\\end{figure}\n\n\\begin{constr}\\label{constr}\n\tIn the remaining part of the article we will use the following geometric construction of the surface $S_{p_{1},p_{2}}$, or rather of its log smooth completion. Let $(x_1,x_2)$ be coordinates on $\\P^{1}\\times \\P^{1}$. For $t\\in \\P^{1}$, $j\\in \\{1,2\\}$ put $\\ll_{j,t}=\\{x_{j}=t\\}$. Write $p_1(t)=t^{d-1}+a_{1}t^{d-2}+\\dots+a_{d-1}$. We perform a sequence of $d$ blowups  over $(0,1)$, as follows.  The exceptional curve of the first blowup is parametrized by $x_{2}'=(x_2-1)x_1^{-1}\\in \\P^{1}$, where $x_{2}'=\\infty$ lies on the proper transform of $\\ll_{1,0}$. We blow up the point $x_{2}'=a_{1}$, and so on: after $m$-th blowup, the exceptional curve is parameterized by $x_{2}^{(m)}=x_2 x_1^{-m}-x_1^{-m}-a_{1}x_1^{-m+1}-\\dots-a_{m-1}$, and we blow up the point with $x_{2}^{(m)}=a_{m}$. Next, we use $p_2$ to analogously define a sequence of $\\deg p_2+1$ blowups over the preimage of $(1,0)$. We denote the resulting morphism by $\\phi\\colon X\\to \\P^{1}\\times \\P^{1}$, put $L_{j,t}:= \\phi^{-1}_{*}\\ll_{j,t}$ and denote by  $A_{1},A_{2}\\subseteq X$ the exceptional curves of the last blowup over $(0,1)$ and $(1,0)$, respectively. Eventually, we put $D=\\phi^{*}(\\ll_{1,\\infty}+\\ll_{2,\\infty}+\\ll_{1,0}+\\ll_{2,0})_{\\mathrm{red}}-A_{1}-A_{2}$ and $S_{p_1,p_2}=X\\setminus D$.\n\t\n\tFigure \\ref{fig:Skl} shows the graph of $D+A_{1}+A_{2}$, where we use the following notation: each line denotes a curve isomorphic to $\\P^{1}$ of specified self-intersection number; the solid ones are the components of $D$, the dashed ones are $A_{1}$ and $A_{2}$; and $[(2)_{m}]$ is short for a chain of $m$ $(-2)$-curves (see Section \\ref{sec:log_surfaces}). \n\t\n\tIf $p_1,p_2\\neq 1$ or $p_1=p_2=1$, then $(X,D)$ is a \\emph{standard} completion of $S_{p_1,p_2}$ in the sense of \\cite{FKZ-weighted-graphs}, see Section \\ref{sec:standard}. To obtain a standard completion in the remaining case, blow up at the preimage of the point $(\\infty,\\infty)$ and contract $L_{1,\\infty}+L_{2,0}$: this gives a graph as in Figure \\ref{fig:Sk0}. In Corollary \\ref{cor:st}\\ref{item:distinct} we use \\cite[Corollary 3.36]{FKZ-weighted-graphs} to prove that $S_{p_1,p_2}\\not\\cong S_{p_1',p_2'}$ for $\\{p_1,p_2\\}\\neq \\{p_1',p_2'\\}$ just by comparing these graphs.\n\\end{constr}\n\nIn Section \\ref{sec:diff} we describe the diffeomorphism types of surfaces in $\\mathcal{S}_{0}$. The result is summarized in Theorem \\ref{thm:diff}. We refer to \\cite[\\S 4,5]{GS_Kirby} for a language of Kirby diagrams (see also Section \\ref{sec:Kirby_overview}) and use the notation of \\cite{BW_surgery} for $2$-bridge knots.\n\n\\begin{thm}\\label{thm:diff}\n\tThe surface $S_{p_{1},p_{2}}$ from Theorem \\ref{THM} is diffeomorphic to the interior of a $2$-handlebody on a $0$-framed $2$-bridge knot $K_{[2d_1,2d_2]}$, where $d_{j}=\\deg p_{j}+1$, $j\\in \\{1,2\\}$. \n\tMoreover, $S_{p_{1},p_{2}}$ is not homeomorphic to $S_{p_{1}',p_{2}'}$ for $\\{\\deg p_{1},\\deg p_{2}\\}\\neq \\{\\deg p_{1}', \\deg p_{2}'\\}$.  \n\\begin{figure}[ht]\n\t\\includegraphics[scale=0.35]{figures\/knot}\n\t\\caption{Kirby diagram of a $4$-manifold whose interior is $S_{p_{1},p_{2}}$ with $\\deg p_{j}=d_{j}-1$.}\n\t\\label{fig:knot}\n\\end{figure}\t\n\\end{thm}\nTheorem \\ref{thm:diff} follows by some elementary handle slides from Proposition \\ref{prop:handles}, where we translate Construction \\ref{constr} to the Kirby diagram in Figure \\ref{fig:handles}. \n\t \t \\begin{figure}[ht]\n\t \t \t\\includegraphics[scale=0.4]{figures\/handles}\n\t \t \t\\caption{Kirby diagram equivalent to the one in Figure \\ref{fig:knot}, cf.\\ \\cite[Fig.\\ 5.14]{GS_Kirby}.}\n\t \t \t\\label{fig:handles}\n\t \t \\end{figure}\t\n\t \t \nIn Corollary \\ref{cor:not_homeo} we prove that $S_{p_{1},p_{2}}$ for different degrees of $p_{j}$ are not homeomorphic, because they have different homotopy types at infinity, see \\cite[1.21]{Fujita-noncomplete_surfaces}. Like in the proof of Theorem \\ref{THM}, we deduce it from the graph of $D$, this time applying the calculus of graph $3$-manifolds \\cite{Neumann-plumbing_graphs} to $M=\\d \\mathrm{Tub}(D)$, see \\cite{Mumford-surface_singularities}. That argument becomes more explicit once it is related to the fact that, by Theorem \\ref{thm:diff}, $M$ is a $0$-surgery on $K_{[2d_1,2d_2]}$.  We exploit it in Section \\ref{sec:pi_1}, giving a thorough geometric description of $M$. The fundamental group $\\pi_{1}(M)=\\pi_{1}^{\\infty}(S_{p_{1},p_{2}})$ is computed in Proposition \\ref{prop:pi_1}.\t\n\n\\begin{rem}\n\tWe do not know which of the $S_{p_{1},p_{2}}$, for fixed degrees of $p_{1},p_{2}$, are biholomorphic. \n\\end{rem}\n\nAn interesting homogeneity property of $S_{1,1}$ was shown in \\cite[\\S 8]{Kal-Ku1_AL-theory}: although $S_{1,1}$ has no algebraic automorphisms except $(x_1,x_2)\\mapsto(x_2,x_1)$, it admits a lot of nice holomorphic ones, coming from the flows of algebraic vector fields. Theorem \\ref{thm:AAut} shows that the same holds for all $S_{p_{1},p_{2}}$. Denote by $\\mathrm{AAut}_{\\mathrm{hol}}(S_{p_1,p_2})$ the subgroup of $\\operatorname{Aut}_{\\mathrm{hol}}(S_{p_1,p_2})$ generated by elements of flows of complete algebraic vector fields on $S_{p_{1},p_{2}}$. \n\n\\begin{thm}\\label{thm:AAut}\n\tWe keep notation from Construction \\ref{constr}.\n\t\\begin{enumerate}\n\t\t\\item\\label{item:Aut_thm} If $p_1\\neq p_2$ then $\\operatorname{Aut}(S_{p_1,p_2})$ is trivial. Otherwise, $\\operatorname{Aut}(S_{p,p})=\\mathbb{Z}_{2}$ is generated by $(x_1,x_2)\\mapsto (x_2,x_1)$.\n\t\t\\item\\label{item:AAut_thm} The group $\\mathrm{AAut}_{\\mathrm{hol}}(S_{p_{1},p_{2}})$ acts $m$-transitively, for any $m$, on its open orbit $S_{p_1,p_2}\\setminus Z$, where:\n\t\t\\begin{enumerate}\n\t\t\t\\item $Z=\\emptyset$ if $p_{1}=p_{2}=1$,\n\t\t\t\\item $Z=((L_{3-j,1}\\cap A_{j}) \\cup A_{3-j})\\cap S_{p_1,p_2} \\cong \\{\\mathrm{pt}\\}\\sqcup \\mathbb{C}^{1}$ if $p_{j}=1, p_{3-j}\\neq 1$ for some $j\\in \\{1,2\\}$,\n\t\t\t\\item $Z=(A_{1}\\cup A_{2})\\cap S_{p_1,p_2}\\cong \\mathbb{C}^{1}\\sqcup \\mathbb{C}^{1}$ if $p_{1},p_{2}\\neq 1$.\n\t\t\\end{enumerate}\n\t\\end{enumerate}\n\\end{thm}\n\nPart \\ref{item:Aut_thm} is obtained as  Corollary \\ref{cor:st}\\ref{item:Aut} from the uniqueness of standard boundary. To prove \\ref{item:AAut_thm} it suffices, by \\cite[Theorem 1.6]{KKL_avf}, to describe all $\\mathbb{C}^{*}$-fibrations of $S_{p_{1},p_{2}}$: this is done in Proposition \\ref{prop:fibrations}. \n\\bigskip\n\n\nWe now explain what is missing in the proof of \\cite[Theorem 2]{GM_k<2}, and how we are going to correct it. The aforementioned result is  deduced from the (partial, but sufficient) classification of almost minimal pairs of Kodaira dimension zero given in \\cite[\\S 8]{Fujita-noncomplete_surfaces}. For the general statement see \\cite{Kojima_k0}. \n\nMore precisely, let $S\\in \\mathcal{S}_{0}$ and let $(X_0,D_0)$ be a smooth completion of $S$. Then by the Miyanishi's theory of peeling (see \\cite[p.\\ 107]{Miyan-OpenSurf} or Section \\ref{sec:peeling} below), there is a sequence of birational morphisms\n\\begin{equation*}\n(X_0,D_0)\\toin{\\psi_{1}}(X_{1},D_{1})\\toin{\\psi_{2}}\\dots \\toin{\\psi_{n}} (X_{n},D_{n}),\n\\end{equation*}\nsuch that $(X_{n},D_{n})$ is almost minimal, hence known, and each $\\psi_{i}$ is an snc-minimalization of $D_{i-1}+A$, where $A\\subseteq X_{i-1}$ is a $(-1)$-curve not contained in $D_{i-1}$ such that $A\\cdot D_{i-1}=1$. The latter process is called a \\emph{half-point attachment} in \\cite[8.15]{Fujita-noncomplete_surfaces}. Therefore, to find $S$ it suffices to identify an $(X_{n},D_{n})$ on the list in \\cite[8.70]{Fujita-noncomplete_surfaces} (cf.\\  \\cite[Proposition 1.5]{Kojima_k0}), and perform suitable half-point attachments.\n\nThe proof in \\cite[Theorem 2]{GM_k<2} proceeds exactly along these lines. However, what is meant in loc.\\ cit.\\ by a half-point attachment is just one blowup, the contraction of $A$ (cf.\\ e.g.\\ \\cite[Definition 4.15]{FZ-deformations}). Nonetheless, if $A$ meets a $(-2)$-curve $C\\subseteq D_{i-1}$ which is not branching in $D_{i-1}$, then after the contraction of $A$, the image of $D_{i-1}$ is not snc-minimal, and $\\psi_{i}$ should contract the image of $C$, too. \n\nThis is why in \\cite{GM_k<2} only $S_{1,1}$ is reconstructed. Indeed, it is shown (essentially) that $n=2$, $X_{2}=\\P^{1}\\times \\P^{1}$ and $D_{2}$ is a union of two vertical and two horizontal lines. But $(X,D)\\to (X_{2},D_{2})$ is obtained by two blowups at smooth points of $D_2$, see Figure \\ref{fig:S00}. The smooth completions of $S_{p_1,p_2}$, which are obtained from $(X_{2},D_{2})$ by two \\emph{sequences of blowups} over $D_{2}$, see Figure \\ref{fig:Skl}, are missing.\n\nIn Section \\ref{sec:proof} we repeat the above proof, filling in the missing part. Our approach contains no essentially new ingredient. However, since the assumption $S\\in \\mathcal{S}_{0}$ is substantially stronger than just $\\kappa(S)=0$, we do not need to rely on the structure theorem \\cite[8.70]{Fujita-noncomplete_surfaces}, as it is done in \\cite{GM_k<2}. So for the convenience of the reader, we make our proof more  self-contained. The only general result we refer to is the theorem of Kawamata \\cite[2.2]{Kawamata-classification_surfaces} which asserts that if $\\kappa(S)=0$ then the positive part of the Zariski decomposition of $K_{X_{n}}+D_{n}$ is numerically trivial, see \\cite[II.6.2.1]{Miyan-OpenSurf}. \n\\bigskip\n\n\\textbf{Acknowledgments.} We would like to thank the referee for valuable comments, including pointing out a flaw in the earlier version of the proof. The first author would like to thank prof.\\ Frank Kutzschebauch for drawing his attention to the article \\cite{FKN_k0} and for many insightful discussions on the subject.\n\n\n\\section{Preliminaries}\\label{sec:prelims}\n\n\\@startsection{subsection}{3{Log surfaces}\\label{sec:log_surfaces}\n\nWe now briefly recall the language of log surfaces. For a complete introduction we refer to \\cite{Fujita-noncomplete_surfaces}.\n\nBy a \\emph{curve} we mean an irreducible, reduced variety of dimension $1$.\n\nLet $X$ be a smooth projective surface. A curve $C\\subseteq X$ is called an \\emph{$n$-curve} if $C\\cong \\P^{1}$ and $C^{2}=n$. In particular, a $(-1)$-curve is an exceptional curve of a blowup, and a $0$-curve is a fiber of a $\\P^{1}$-fibration.\n\nLet $D$ be an effective $\\mathbb{Q}$-divisor on $X$. By a \\emph{component} of $D$ we always mean an irreducible component. The number of components of $D$ is denoted by $\\#D$. The \\emph{branching number} of a component $C\\subseteq D$ is\n\\begin{equation*}\n\\beta_{D}(C)=C\\cdot (D-C).\n\\end{equation*}\nWe say that $C$ is a \\emph{tip} of $D$ if $\\beta_{D_{\\mathrm{red}}}(C)=1$, and \\emph{branching} in $D$  if $\\beta_{D_{\\mathrm{red}}}(C)\\geq 3$. \n\nAssume that $D$ is \\emph{simple normal crossing} (\\emph{snc}), that is, $D$ is reduced, all components of $D$ are smooth and meet transversally, at most two at each point. A $(-1)$-curve $C\\subseteq D$ is called \\emph{superfluous} if $1\\leq \\beta_{D}(C)\\leq 2$ and $C$ meets each component of $D$ at most once: in other words, after contraction of $C$, the image of $D$ remains snc. We say that $D$ is \\emph{snc-minimal} if it contains no superfluous $(-1)$-curves. An \\emph{snc-minimalization} of $D$ is the contraction of all superfluous $(-1)$-curves in $D$ and its images.\n\nAssume further that $D$ has connected support and $\\beta_{D}(C)\\leq 2$ for all components $C$ of $D$. We say that $D$ is a \\emph{chain} if at least one inequality is strict, otherwise we say that $D$ is \\emph{circular}. We order the components $T_{1},\\dots, T_{m}$ of $D$ in such a way that $T_{i}\\cdot T_{i+1}=1$ for $i\\in \\{1,\\dots, m-1\\}$, and $T_{1}$ is a tip of $D$ in case $D$ is a chain. The sequence of integers $(-T_{1}^{2},\\dots, -T_{m}^{2})$ is then called a \\emph{type} of $D$. We often abuse notation and write \n\\begin{equation*}\nD=[-T_{1}^{2},\\dots,-T_{m}^{2}]\\mbox{ if $D$ is a chain and } D=(\\!( -T_{1}^{2},\\dots, -T_{m}^{2})\\!) \\mbox{ if $D$ is circular}.\n\\end{equation*}\nA sequence consisting of an integer $a$ repeated $k$ times will be abbreviated as $(a)_{k}$.\n\nAn effective divisor $D$ is called a \\emph{tree} if $D_{\\mathrm{red}}$ is snc and has no circular subdivisor. A \\emph{fork} is a tree with exactly one branching component and three maximal twigs. A \\emph{rational} tree (chain, fork...) is a tree (chain,  fork...) whose components are all rational.\n\n\nWe say that a reduced divisor $D=\\sum D_{i}$ is \\emph{negative definite} if its intersection matrix $[D_{i}\\cdot D_{j}]_{1\\leq i,j\\leq r}$ is.\n\nLet again $D$ be an snc divisor. A chain $T\\subseteq D$ is a \\emph{twig} of $D$ if it contains a tip of $D$ and its components are non-branching in $D$. We order a twig in such a way that its first component is a tip of $D$. A twig is called \\emph{admissible} if all its components are rational and have self-intersection number at most $-2$. An (admissible) twig of $D$ is \\emph{maximal} if it is maximal in the set of (admissible) twigs ordered by inclusion of supports. A \\emph{$(-2)$-twig} is a twig whose components are $(-2)$-curves.\n\nLet $T$ be an admissible twig of $D$. Then $T$ is negative definite, so there is a unique $\\mathbb{Q}$-divisor $\\operatorname{Bk}_{D}(T)\\leq T$, called the \\emph{bark} of $T$,  such that for every component $T_{0}$ of $T$ \n\\begin{equation*}\nT_{0}\\cdot\\operatorname{Bk}_{D}(T)=T_{0}\\cdot (K_{X}+D),\n\\end{equation*}\nthat is, $T_{0}\\cdot \\operatorname{Bk}_{D}(T)=-1$ if $T_{0}$ is a tip of $D$ and $T_{0}\\cdot \\operatorname{Bk}_{D}(T)=0$ otherwise. An important result for the theory of peeling asserts that the coefficients of $\\operatorname{Bk}_{D}(T)$ are strictly between $0$ and $1$  \\cite[II.3.3]{Miyan-OpenSurf}.\n\nAssume that $D$ is a connected snc-minimal divisor which is not negative definite. Then $\\operatorname{Bk} D$ is defined as the sum of barks of all maximal admissible twigs of $D$. We put $D^{\\#}=D-\\operatorname{Bk} D$.\n\\smallskip\n\nLet $D$ be an snc divisor on $X$. We say that a blowup $X'\\to X$ at a point $p\\in D$ is \\emph{outer} with respect to $D$ if $p$ is a smooth point of $D$, otherwise it is called \\emph{inner}. We say that a birational map $X\\dashrightarrow Y$ \\emph{touches} a divisor if it is not an isomorphism in any of its neighborhoods.\n\\smallskip\n\n\nA \\emph{log smooth} pair $(X,D)$ consists of a smooth projective surface $X$ and an snc divisor $D$ on $X$. Any smooth surface $S$ admits a \\emph{log smooth completion}, that is, a log smooth pair $(X,D)$ such that $S\\cong X\\setminus D$. We say that such a completion is \\emph{minimal} if it does not dominate birationally any other log smooth completion, or, equivalently, if $D$ is snc-minimal.\n\n\\@startsection{subsection}{3{Affine surfaces whose coordinate rings are factorial and have trivial units}\n\n\nFor a divisor $D$ on a smooth surface $X$, we denote by $\\mathbb{Z}[D]$ the free abelian group whose basis is the set of components of $D$. There is a natural group homomorphism $\\mathbb{Z}[D]\\to\\operatorname{Pic}(X)$. The following lemma is well known, cf.\\ \\cite[1.17]{Fujita-noncomplete_surfaces} or \\cite[Lemma 2.2]{GM_k<2}. \n\n\\begin{lem}[Criterion for $S\\in \\mathcal{S}_{\\kappa}$]\\label{lem:S0}\n\tLet $(X,D)$ be a smooth completion of a smooth affine surface $S$.\n\t\\begin{enumerate}\n\t\t\\item\\label{item:UFD} $\\mathbb{C}[S]$ is a UFD if and only if $\\mathbb{Z}[D]\\to\\operatorname{Pic}(X)$ is surjective.\n\t\t\\item\\label{item:units} $\\mathbb{C}[S]^{*}=\\mathbb{C}^{*}$ if and only if $\\mathbb{Z}[D]\\to\\operatorname{Pic}(X)$ is injective.\n\t\\end{enumerate}\n\tAssume that $S\\in \\mathcal{S}_{\\kappa}$. Then $\\mathbb{Z}[D]=\\operatorname{Pic}(X)=\\operatorname{NS}(X)$ and $q(X)=0$. If $\\kappa<2$ then $X$ is rational.\n\\end{lem}\n\\begin{proof}\n\t\\ref{item:UFD} By \\cite[II.6.2]{Hartshorne_AG}, $\\mathbb{C}[S]$ is factorial if and only if the divisor class group $\\mathrm{Cl}(S)$ of $S$ is trivial. It follows from \\cite[II.6.5(c)]{Hartshorne_AG} that $\\mathrm{Cl}(S)$ is the cokernel of the natural map $\\mathbb{Z}[D]\\to\\mathrm{Cl}(X)$. Eventually, $\\mathrm{Cl}(X)=\\operatorname{Pic}(X)$ because $X$ is smooth \\cite[II.6.16]{Hartshorne_AG}.\n\t\n\t\\ref{item:units} A regular function on $S$ is invertible if and only if it extends to $f\\in \\mathbb{C}(X)$ with all zeros and poles in $D$, that is, with $\\mathrm{div}(f)\\in \\mathbb{Z}[D]$. Two rational functions on $X$ with the same divisor differ by an element of $\\O_{X}(X)^{*}=\\mathbb{C}^{*}$, so $f\\mapsto \\mathrm{div}(f)$ gives an isomorphism of $\\mathbb{C}[S]^{*}\/\\mathbb{C}^{*}$ with the kernel of $\\mathbb{Z}[D]\\to \\operatorname{Pic}(X)$.\n\t\n\tFor the remaining part, we follow \\cite[Lemma 2.2]{GM_k<2}. Assume that $S\\in \\mathcal{S}_{\\kappa}$. Then \\ref{item:UFD}, \\ref{item:units} give $\\mathbb{Z}[D]=\\operatorname{Pic}(X)$. The exponential sequence induces an exact sequence\n\t\\begin{equation*}\n\t0\\to H^{1}(X,\\mathbb{Z})\\to H^{1}(X,\\O_{X})\\to \\operatorname{Pic}(X)\\to H^{2}(X,\\mathbb{Z}).\n\t\\end{equation*}\n\tSince $H^{1}(X,\\O_{X})$ is a vector space over $\\mathbb{C}$, and by \\ref{item:UFD} $\\operatorname{Pic}(X)$ is a finitely generated abelian group, we obtain $q(X)=h^{1}(X,\\O_{X})=0$ and that $\\operatorname{Pic}(X)\\to H^{2}(X,\\mathbb{Z})$ is injective, i.e.\\ $\\operatorname{Pic}(X)=\\operatorname{NS}(X)$. Now to prove that $X$ is rational, it remains to show that $p_{g}(X)=0$ \\cite[IV.6.2]{Hartshorne_AG}. Assume $\\kappa<2$. Because $S$ is affine, $D$ supports an  effective ample divisor, so if $|K_{X}|\\neq \\emptyset$ then for $m\\gg 0$ we have $h^{0}(m(K_{X}+D))\\sim m^{2}$, hence $\\kappa=2$; a contradiction.\n\\end{proof}\n\n\\begin{rem}\\label{rem:open_UFD}\n\tLemma \\ref{lem:S0}\\ref{item:UFD} implies that an open affine subset of a smooth factorial surface is factorial, and that the converse holds if and only if its complement is a principal divisor. This is a special case of the Nagata lemma, see e.g.\\ \\cite[Lemma 19.20]{Eisenbud_comm-alg}\n\\end{rem}\n\n\\begin{cor}[$S_{p_1,p_2}\\in \\mathcal{S}_{\\kappa}$]\\label{cor:S_k}\n\tLet $(X,D)$ be the log smooth completion of $S_{p_1,p_2}$ as in Construction \\ref{constr}. Then $\\operatorname{Pic}(X)=\\mathbb{Z}[D]$. In particular,  $\\mathbb{C}[S_{p_1,p_2}]$ is a UFD and $\\mathbb{C}[S_{p_1,p_2}]=\\mathbb{C}^{*}$.\n\\end{cor}\n\\begin{proof}\n\tWe use the notation introduced in Construction \\ref{constr}. The group $\\operatorname{Pic}(X)$ is generated by the components of $\\phi^{*}(\\ll_{1,\\infty}+\\ll_{1,0}+\\ll_{2,\\infty}+\\ll_{2,0})_{\\mathrm{red}}=D+A_{1}+A_{2}$, with relations $0=\\phi^{*}(\\ll_{j,0}-\\ll_{j,\\infty})=A_{j}+T_{j}+L_{j,0}-L_{j,\\infty}$, $j\\in \\{1,2\\}$, where $T_{j}$ is the twig of $D$ meeting $L_{j,0}$. Therefore, $\\operatorname{Pic}(X)=\\mathbb{Z}[D+A_{1}+A_{2}]\/\\mathbb{Z}[A_{1}+A_{2}]=\\mathbb{Z}[D]$. The remaining assertion follows from Lemma \\ref{lem:S0}.\n\\end{proof}\n\n\\@startsection{subsection}{3{$\\P^{1}$-fibrations}\n\nA \\emph{fibration} of a smooth surface $S$ is a surjective morphism onto a curve whose general fiber is irreducible and reduced.  A curve on $S$ is called \\emph{vertical} (resp.\\ \\emph{horizontal}) if $C\\cdot F=0$ (resp.\\ $C\\cdot F>0$) for a fiber $F$; it is called an \\emph{$n$-section} if $C\\cdot F=n$. Any divisor $D$ decomposes uniquely as $D=D\\vert+D_{\\mathrm{hor}}$, where all components of $D\\vert$ are vertical and all components of $D_{\\mathrm{hor}}$ are horizontal.\n\nA $\\P^{1}$- (resp.\\ $\\mathbb{C}^{1}$-, $\\mathbb{C}^{*}$-) fibration is a fibration whose general fiber is isomorphic to $\\P^{1}$ (resp.\\ $\\mathbb{C}^{1}$, $\\mathbb{C}^{*}:= \\mathbb{C}^{1}\\setminus \\{0\\}$). A fiber not isomorphic to a general one will be called \\emph{degenerate}. \n\nThe surfaces in $\\mathcal{S}_{0}$ do not admit $\\mathbb{C}^{1}$-fibrations by the Iitaka Easy Addition Theorem \\cite[Theorem 11.9]{Iitaka_AG}, but they admit plenty of $\\mathbb{C}^{*}$-fibrations (eg.\\ produced by Lemma \\ref{lem:producing_Cst}\\ref{item:HP_Cst}), which we will use in Proposition \\ref{prop:fibrations}\\ref{item:vf} as first integrals for complete algebraic vector fields, see \\cite[Theorem 1.6]{KKL_avf}. The structure of $\\mathbb{C}^{*}$-fibrations is well described in \\cite[\\S 7]{Fujita-noncomplete_surfaces}, however, for our purposes it will be more convenient to study them directly, by completing them to $\\P^{1}$-fibrations. \n\nLet $F$ be a degenerate fiber of a $\\P^{1}$-fibration of a smooth projective surface. Then $F$ is obtained from a $0$-curve by a sequence of blowups, hence its geometry is easy to understand, see eg.\\ \\cite[\\S 4]{Fujita-noncomplete_surfaces}. In particular, $F$ is a rational tree with no branching $(-1)$-curves; moreover, if a $(-1)$-curve has multiplicity $1$ in $F$ then it is a tip of $F$, and $F$ contains another $(-1)$-curve \\cite[7.3]{Fujita-noncomplete_surfaces}.\n\nThe following observation, see \\cite[7.14(1)]{Fujita-noncomplete_surfaces}, will be used several times.\n\n\\begin{lem}[our $\\mathbb{C}^{*}$-fibrations are untwisted]\\label{lem:untwisted}\n\tLet $(X,D)$ be a log smooth completion of a smooth affine surface $S$ such that $\\mathbb{C}[S]$ is a UFD. Let $H_{1},\\dots, H_{r}$ be all components of $D_{\\mathrm{hor}}$ for some $\\P^{1}$-fibration of $X$. Put $h_j=H_{j}\\cdot F$ for a fiber $F$. Then $\\gcd(h_{1},\\dots, h_{r})=1$.\n\\end{lem}\n\\begin{proof}\n\tBy Tsen's theorem, $X$ contains a $1$-section $H$. By Lemma \\ref{lem:S0}\\ref{item:UFD}, $H\\equiv V+\\sum_{j=1}^{r} a_{j}H_{j}$ for some vertical $V$ and $a_{1},\\dots, a_{r}\\in \\mathbb{Z}$. Intersecting with $F$ gives $1=\\sum_{j=1}^{r} a_{j}h_{j}$, so $\\gcd(h_1,\\dots, h_r)=1$.\n\\end{proof}\n\n\n\n\n\n\\@startsection{subsection}{3{Standard completions and elementary transformations}\\label{sec:standard}\n\nA minimal log smooth completion of an open surface may not be unique. However, \\cite[Definition 2.13]{FKZ-weighted-graphs} distinguishes a class of \\emph{standard completions}, which gives a convenient tool to tell affine surfaces apart. We now recall this definition. A rational chain is \\emph{standard} if it is of type\n\\begin{equation*}\n[(0)_{2k+1}]\\quad\\mbox{or}\\quad[(0)_{2k},a_{1},\\dots, a_{l}]\\quad\\mbox{for some integers }k,l\\geq 0\\mbox{ and }a_{1},\\dots,a_{l}\\geq 2.\n\\end{equation*}\nA rational circular divisor is \\emph{standard} if it is of type\n\\begin{equation*}\n(\\!((0)_{2k},a_{1},\\dots, a_{l})\\!)\\mbox{ or }\n(\\!((0)_{k},a)\\!)\\mbox{ or }\n(\\!((0)_{2k},1,1)\\!)\\quad\\mbox{for some }k,l\\geq 0,\\ a\\geq 0\\mbox{ and }a,a_{1},\\dots,a_{l}\\geq 2.\n\\end{equation*}\nLet now $D$ be any snc divisor, and let $B$ be the sum of all components of $D$ which are either branching or non-rational; so every connected component of  $D-B$ is rational, chain or circular. We say that $D$ is standard if every connected component of $D-B$ is standard in the above sense. A log smooth completion $(X,D)$ is \\emph{standard} if $D$ is. Note that the log smooth completions of $S_{p_1,p_2}$ in Figure \\ref{fig:boundaries} are standard.\n\\smallskip\n\nLet $D$ be an snc divisor on $X$, and let $C\\subseteq D$ be a non-branching $0$-curve. If $\\beta_{D}(C)=2$ choose $p\\in C\\cap (D-C)$, otherwise let $p$ be any point of $C$. A \\emph{flow} (or \\emph{elementary transformation}) \\emph{on} $C$ is the birational map $(X,D)\\dashrightarrow (X',D')$, where $D'$ is the reduced total transform of $D$, defined as a blowup at $p$, followed by the contraction of the proper transform of $C$. We remark that in Section \\ref{sec:AAut} we will use the word \\enquote{flow} in its standard meaning, i.e.\\ flow of a vector field: this will lead to no confusion.\n\nLet $\\phi\\colon (X,D)\\dashrightarrow (X',D')$ be a flow on $C\\subseteq D$. Then  $\\phi|_{X\\setminus D}\\colon X\\setminus D\\to X'\\setminus D'$ is an isomorphism. In particular, if the components of $D$ generate $\\operatorname{Pic}(X)$ (respectively, are $\\mathbb{Z}$-linearly independent in $\\operatorname{Pic}(X)$), then by Lemma \\ref{lem:S0} the same is true for the components of $D'$ in $\\operatorname{Pic}(X')$. Moreover, $\\phi$ does not touch components of $D$ other than $C$ and the components of $D$ meeting $C$. If $C$ is not a tip of $D$, then $\\phi$ replaces a subchain $[a,0,b]\\subseteq D$, whose middle component is $C$, by $[a+1,0,b-1]\\subseteq D'$.\n\\smallskip\n\nWe are ready to formulate the result used to distinguish the surfaces $S_{p_1,p_2}$ for different $\\{p_1,p_2\\}$.\n\n\\begin{lem}[{\\cite[Corollary 3.36]{FKZ-weighted-graphs}}]\\label{lem:boundaries}\n\tAny smooth affine surface admits a standard log smooth completion. Any two such completions differ by a sequence of flows on some $0$-curves in the boundary.\n\\end{lem}\n\n\\begin{cor}[$S_{p_1,p_2}$ are non-isomorphic]\\label{cor:st}\n\tLet $S_{p_1,p_2}$ be as in Theorem \\ref{THM}. Then\n\t\\begin{enumerate}\n\t\t\\item\\label{item:st_uniq} The standard completion $(X,D)$ of $S_{p_1,p_2}$ is unique up to an isomorphism, with $D$ as in Figure \\ref{fig:boundaries}.\n\t\t\\item\\label{item:distinct} The surfaces $S_{p_1,p_2}$ and $S_{p_1',p_2'}$ are not isomorphic unless $\\{p_1,p_2\\}=\\{p_1',p_2'\\}$.\n\t\t\\item \\label{item:Aut} If $p_1\\neq p_2$ then $\\operatorname{Aut}(S_{p_1,p_2})$ is trivial. Otherwise, $\\operatorname{Aut}(S_{p,p})=\\mathbb{Z}_{2}$ is generated by $(x_1,x_2)\\mapsto (x_2,x_1)$.\n\t\\end{enumerate}\n\\end{cor}\n\\begin{proof}\n\t\\ref{item:st_uniq} By Lemma \\ref{lem:boundaries}, $D$ differs from the one in Figure \\ref{fig:boundaries} by a sequence of flows. Because all twigs of the one in Figure \\ref{fig:boundaries} are admissible, they are not touched by any flow. Since a flow preserves the non-weighted graph and the sum of self-intersection numbers; we only need to check that the latter agree.\n\t\n\tConsider the case when $p_1,p_2\\neq 1$ or $p_1=p_2=1$. Then any such $D$ contains a rational circular divisor $L$ with components, say, $L_{1},L_{2},L_{3},L_{4}$, such that $L_{i}\\cdot L_{i+1}=1$ for $i\\in \\{1,2,3\\}$; if $p_{j}\\neq 1$ for $j\\in \\{1,2\\}$ then $L_{2+j}$ meets a twig of $D$ of type $[(2)_{\\deg p_j}]$, and $L_{1}^{2}=0$, i.e.\\ $L_{1}$ is the $0$-curve on which the last flow is performed. If $p_1,p_2\\neq 1$ then the chain $L_{1}+L_{2}$ is standard, and if $p_1=p_2=1$ then the whole $L$ is standard. In any case, we infer that $L_{2}^{2}=0$. It follows that $L_{3}^{2},L_{4}^{2}<0$: indeed, for $i\\in \\{1,2\\}$, $L_{i+2}$ is vertical for the $\\P^{1}$-fibration of $X$ induced by $|L_{i}|$, and it is not a fiber because otherwise $L_{i+2}=L_{i}$ in $\\operatorname{Pic}(X)$, contrary to Lemma \\ref{lem:S0}\\ref{item:units}. Because the sum $L_{1}^{2}+L_{2}^{2}+L_{3}^{2}+L_{4}^{2}$ is not changed by a flow, we get $L_{3}^{2}+L_{4}^{2}=-2$, so $L_{3}^{2}=L_{4}^{2}=-1$, as claimed.\n\t\n\tIn case $p_j\\neq 1$, $p_{3-j}= 1$ we similarly obtain that $D$ has a rational circular divisor $L_{1}+L_{2}+L_{3}$, where $L_{3}$ meets a twig $[(2)_{\\deg p_j}]$ and $L_{1}$ is a $0$-curve. The chain $L_{1}+L_{2}$ is standard, so $L_{2}^{2}=0$, and $L_{3}^{2}=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}=1$.\n\t\n\t\\ref{item:distinct}, \\ref{item:Aut} Let $S_{p_1,p_2}\\to S_{p_1',p_2'}$ be an isomorphism. By Lemma \\ref{lem:boundaries}, it extends to an isomorphism, say $\\tau\\colon (X_1,D_1)\\to(X_2,D_2)$, between their standard completions, which by \\ref{item:st_uniq} are both as in Construction \\ref{constr}. If they are as in Figure \\ref{fig:Sk0}, we reverse the last step of Construction \\ref{constr} and lift $\\tau$ accordingly, so that $(X_{i},D_{i})$ are as in Figure \\ref{fig:Skl} or \\ref{fig:S00} (but possibly no longer standard). \n\t\n\tFor $i,j\\in \\{1,2\\}$ let $A_{j,i}\\subseteq X_{i}$ be as in Figure \\ref{fig:boundaries}. Because by Lemma \\ref{lem:S0} $\\operatorname{Pic}(X_{i})=\\mathbb{Z}[D_{i}]$, the curve $\\tau(A_{j,1})$ is linearly equivalent to $A_{j,2}$. In fact, $\\tau(A_{j,1})=A_{j,2}$ because $\\tau(A_{j,1})\\cdot A_{j,2}=A_{j,2}^{2}<0$. Therefore, $\\tau$ descends to an automorphism $\\tau_{0}$ of $\\P^{1}\\times \\P^{1}$ fixing $\\ll_{1,0}+\\ll_{1,\\infty}+\\ll_{2,0}+\\ll_{2,\\infty}$ and the points $(0,1)$, $(1,0)$, so  $\\tau_{0}(x_1,x_2)=(x_1,x_2)$ or $(x_2,x_1)$. Hence $\\tau$ or $\\tau\\circ \\epsilon$, where $\\epsilon(x_1,x_2)=(x_2,x_1)$, is trivial on each exceptional curve of $\\phi\\colon X\\to \\P^{1}\\times \\P^{1}$. This shows that $\\{p_1,p_2\\}=\\{p_1',p_2'\\}$ and $\\tau=\\mathrm{id}$ or $\\epsilon$.\n\\end{proof}\n\n\\begin{rem}[{$S_{p_1,p_2}$ are stably non-isomorphic, cf.\\ \\cite[Corollary 4.3]{FKN_k0}}]\n\tBecause $\\kappa(S_{p_1,p_2})=0$ by Corollary \\ref{cor:S_kappa} below, \\cite{Iitaka_Fujita-cancellation}  implies that $S_{p_1,p_2}\\times \\mathbb{C}^{n}\\not\\cong S_{p'_1,p'_2}\\times \\mathbb{C}^{n}$ for any $n\\geq 0$ and $\\{p_1,p_2\\}\\neq \\{p_1',p_2'\\}$.\n\\end{rem}\n\nWe conclude with a well-known application of flows, which will be useful in the proof of Lemma \\ref{lem:producing_Cst}.\n\n\\begin{lem}[{\\cite[6.13]{Fujita-noncomplete_surfaces}, cf.\\ \\cite[Lemma 4.14]{FZ-deformations}}]\\label{lem:rational-twigs-are-admissible}\n\tLet $(X,D)$ be a log smooth pair with $\\kappa(X\\setminus D)\\geq 0$. Let $T$ be a rational twig of $D$. Then $C^2\\leq -1$ for every component $C$ of $T$. In particular, if $D$ is snc-minimal then all rational twigs of $D$ are admissible.\n\\end{lem}\n\\begin{proof}\n\tLet $T_{k}$ be the $k$-th component of $T$, put $T_0=0$. Suppose $T_{k}^{2}\\geq 0$ for some $k\\in \\{1,\\dots, \\#T\\}$. Write $\\{p\\}=T_{k}\\cap (D-T_{k}-T_{k-1})$, or choose $p\\in T_{k}\\setminus T_{k-1}$ if that intersection is empty. Blowing up $T_{k}^{2}$ times at $p$ and its infinitely near points on the proper transforms of $T_{k}$, we can assume that $T_{k}=[0]$. If $k>1$, then after $\\pm T_{k-1}^{2}$ flows on $T_{k}$, we can assume $T_{k-1}=[0]$, too. Thus by induction, we can assume $k=1$, i.e.\\ $T_{1}=[0]$ is a tip of $D$. Then $|T_{1}|$ induces a $\\P^{1}$-fibration of $X$ which restricts to a $\\mathbb{C}^{1}$-fibration of $X\\setminus D$, so $\\kappa(X\\setminus D)=-\\infty$ by the Iitaka Easy Addition theorem; a contradiction.\n\\end{proof}\n\n\\@startsection{subsection}{3{Graph $3$-manifolds and their normal forms}\\label{sec:graph_manifolds}\n\nAn important topological invariant of an affine surface $S$ is its \\emph{fundamental group at infinity},  $\\pi_{1}^{\\infty}(S):=\\varprojlim \\pi_{1}(S\\setminus K)$, where $\\varprojlim$ runs over all compact $K\\subseteq S$. If $(X,D)$ is a log smooth completion of $S$, then $\\pi_{1}^{\\infty}(S)=\\pi_{1}(M)$, where $M$ is the boundary of a nice tubular neighborhood of $D$, constructed in \\cite{Mumford-surface_singularities} as a plumbed manifold whose graph is the dual graph of $D$, see \\cite[4.6.2]{GS_Kirby}. More precisely, $\\mathrm{Tub}(D)$ is a union, taken over components $C$ of $D$, of disk bundles $\\xi_C$ with Euler numbers $C^2$, where for any component $C'$ of $D$ meeting $C$, a fiber of $\\xi_C$ over $C\\cap C'$ is  glued to a neighborhood of $C\\cap C'$ in $C$.  \n\nFor a general introduction to $3$-manifolds we refer to \\cite{Hatcher_3M,AFW_3M}. \nWe denote by $\\mathbb{S}^{k}$, $\\mathbb{D}^{k}$ the (real) $k$-dimensional sphere and disk, respectively, and by $\\mathbb{T}^{k}=(\\mathbb{S}^{1})^{k}$ the $k$-dimensional torus. We say that a $3$-manifold is \\emph{prime} if it cannot be written as a connected sum of two $3$-manifolds other than $\\mathbb{S}^{3}$, and \\emph{irreducible} if any embedded $\\mathbb{S}^{2}$ bounds a ball. A prime, oriented $3$-manifold is either irreducible or $\\mathbb{S}^{1}\\times \\mathbb{S}^{2}$.\n\nIf $M$ is prime, and not a lens space (which will be the case for $S=S_{p_{1},p_{2}}$), then $\\pi_{1}(M)$ determines $M$ uniquely, up to a diffeomorphism \\cite[Theorem 2.1.2]{AFW_3M}. Therefore, in this case to distinguish the homeomorphism type of $S$ at infinity, it suffices to describe $\\pi_1(M)$, which, in turn, is given by the graph of $D$, once put in a normal form \\cite[Theorem 4.2]{Neumann-plumbing_graphs}. The definition of normal form is similar to the one of standard boundary, but requires some more flexibility to allow more operations between $3$-manifolds than just blowing up and down. We now recall the definition of graph $3$-manifolds following \\cite{Neumann-plumbing_graphs}. We refer to \\cite{EN_torus-links} for details and relation to the JSJ-decomposition (graph manifolds are exactly those for which the latter consists only of Seifert fibered spaces: we compute it for our $M$ in Proposition \\ref{prop:JSJ}).\n\nA \\emph{graph manifold} is a $3$-manifold which is a union of $\\mathbb{S}^{1}$-bundles over compact surfaces, glued according to some  graph, as follows. An $\\mathbb{S}^{1}$-bundle over $C$ with Euler number $e$ is represented by a vertex of weight $e$ and two additional numbers $g$, $r$, where $g$ is the genus of $C$ ($g<0$ if $C$ is non-orientable)  and $r=b_{0}(\\d C)$. They are skipped whenever both are zero, i.e.\\ $C\\cong \\P^{1}$. An edge between two vertices $v_{1}$, $v_{2}$ represents gluing of a fiber of $v_{1}$ to a small loop around a point in a base of $v_2$, and vice versa. Multiple edges and loops are allowed. An edge is labeled by \\enquote{$+$} if the gluing respects the chosen orientations, and \\enquote{$-$} if it reverses both of them.  Clearly, reversing the sign of all edges adjacent to a single vertex, or of any edge adjacent to a vertex with $g<0$, does not change the manifold. Hence we will label only the edges contained in circular subgraphs with no vertices of $g<0$, see \\cite[p.\\ 304]{Neumann-plumbing_graphs}.\n\nFor example, the boundary $M$ from \\cite{Mumford-surface_singularities}, described above, is a graph manifold associated to the dual graph of $D$, with all edges labeled with \\enquote{$+$}. For such graph, we use the notions of branching numbers, chains, twigs as in Section \\ref{sec:log_surfaces}, a vertex being \\enquote{rational} if $g=r=0$.\n\nA connected, rational graph $\\Gamma$ is \\emph{normal} if the following three conditions hold. First, all non-branching vertices of $\\Gamma$ have weights at most $-2$. Second, if a vertex with $\\beta=3$ meets two twigs of type $[2]$ then $\\Gamma$ is a fork. Third, if $\\Gamma=(\\!( (2)_{k} )\\!)$ for some $k> 0$ then at least two edges are labeled with \\enquote{$-$}. To extend this definition to non-rational graphs, one needs to exclude some additional special cases, see \\cite[\\S 4]{Neumann-plumbing_graphs}. We will not need this extension. In general, a graph is \\emph{normal} if all its connected components are.\n\nFor a graph $3$-manifold $N$, \\cite[Theorem 4.1]{Neumann-plumbing_graphs} gives an algorithm to reduce its graph to a normal form, which we denote by $\\Gamma(N)$. This algorithm uses certain operations R0--R8 defined in Proposition 2.1 loc.\\ cit, which do not change the graph manifold. Among these, R1 is blowing down a vertex with weight $\\pm 1$, and R3 is an \\enquote{absorption} of a cylinder $(\\mathbb{D}^{1}\\times \\mathbb{S}^{1})\\times \\mathbb{S}^{1}$ corresponding to a rational vertex of weight $0$ and $\\beta=2$.  The graph $\\Gamma(-N)$ can be directly computed from $\\Gamma(N)$  using \\cite[Theorem 7.1]{Neumann-plumbing_graphs}.\n\n\nNow we can state the main result of \\cite{Neumann-plumbing_graphs} used to distinguish $\\pi_{1}^{\\infty}(S_{p_1,p_2})$ for different degrees of $p_{j}$.\n\n\\begin{lem}[{\\cite[Theorems 4.2, 4.3]{Neumann-plumbing_graphs}}]\\label{lem:graphs}\n\tLet $N,N'$ be graph $3$-manifolds.\n\t\\begin{enumerate}\n\t\t\\item\\label{item:Gamma}  $\\Gamma(N)=\\Gamma(N')$ if and only if $N$ and $N'$ are orientation-preserving diffeomorphic.\n\t\t\\item\\label{item:irr} If $\\Gamma$ is connected then $N$ is prime.\n\t\\end{enumerate}\n\\end{lem}\n\n\n\t \t \\begin{figure}[ht]\t\n\t\\begin{tabular}{cc}\n\t\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\t\\centering\t \t \t\t\t\n\t\t\t\\includegraphics[scale=0.3]{figures\/g_1}\n\t\t\t\\caption{Graph of $M$ as in \\cite{Mumford-surface_singularities}, see Figure \\ref{fig:Skl}}\n\t\t\t\\label{fig:g_1}\n\t\t\\end{subfigure}\t\n\t\t&\n\t\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\t\\centering\t \t \t\t\t\n\t\t\t\\includegraphics[scale=0.3]{figures\/g_2}\n\t\t\t\\caption{$\\Gamma(M)$ for $p_{1},p_{2}\\neq 1$}\n\t\t\t\\label{fig:g_2}\n\t\t\\end{subfigure}\n\t\t\\\\\n\t\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\t\\centering\t \t \t\t\t\n\t\t\t\\includegraphics[scale=0.35]{figures\/g_3}\n\t\t\t\\caption{$\\Gamma(M)$ for $p_{1}=1$, $p_{2}\\neq 1$}\n\t\t\t\\label{fig:g_3}\n\t\t\\end{subfigure}\t\n\t\t&\n\t\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=0.3]{figures\/g_4}\n\t\t\t\\caption{$\\Gamma(-M)$ for $p_{1},p_{2}=1$, see Proposition \\ref{prop:JSJ}\\ref{item:Moser}}\n\t\t\t\\label{fig:g_4}\n\t\t\\end{subfigure}\n\t\\end{tabular}\n\t\\caption{Graphs of $M=\\d\\mathrm{Tub}(D)$ in the proof of Corollary \\ref{cor:not_homeo}}\n\t\\label{fig:graphs}\n\\end{figure}\n\n\\begin{cor}\\label{cor:not_homeo}\n\tThe surfaces $S_{p_{1},p_{2}}$ and $S_{p_{1}',p_{2}'}$  are not homeomorphic for $\\{\\deg p_{1},\\deg p_{2} \\}\\neq \\{\\deg p_{1}',\\deg p_{2}' \\}$.\n\\end{cor}\n\\begin{proof}\n\tFor $S_{p_{1},p_{2}}$ and $S_{p_{1}',p_{2}'}$, $\\{\\deg p_{1},\\deg p_{2}\\}\\neq\\{\\deg p_{1}',\\deg p_{2}'\\}$ let $M$, $M'$ be a boundary of tubular neighborhoods of respective $D\\subseteq X$ as in Construction \\ref{constr}.  Such $M$ is a graph manifold represented by Figure \\ref{fig:g_1}. Absorption (move R3 in \\cite{Neumann-plumbing_graphs}) of a vertex corresponding to $L_{1,\\infty}$ reduces it to a graph in Figure \\ref{fig:g_2}. It is in normal form if $p_{1},p_{2}\\neq 1$. If, say, $p_{1}=1$ then the normal form is obtained by blowing down (move R1) the vertex corresponding to $L_{1,0}$. If $p_{2}\\neq 1$, then the resulting graph in Figure \\ref{fig:g_3} is normal, otherwise it needs to be replaced by a standard graph of a Seifert fiber space (move R7). In our case this graph is a graph $E_{9}$, which after orientation reversing gives the one in Figure \\ref{fig:g_4}, cf.\\ Proposition \\ref{prop:JSJ}\\ref{item:Moser} and \\cite[p.\\ 309]{Neumann-plumbing_graphs}. \n\t \t \t\n\tHence $\\Gamma(M)\\neq \\Gamma(M')$, so by Lemma \\ref{lem:graphs}\\ref{item:Gamma} $M$ is not orientation-preserving diffeomorphic to $M'$. Reversing orientation in Figures \\ref{fig:graphs}\\subref{fig:g_2}-\\subref{fig:g_3} inverts the edge signs and replaces a twig $[(2)_{d}]$ with $[d+2]$, so the resulting graph is different from the previous ones. Therefore, $M\\not\\cong_{\\mathrm{diffeo}}M'$, so $M\\not\\cong_{\\mathrm{homeo}}M'$, see \\cite[1.1]{AFW_3M}. Because $\\Gamma(M)$ is connected and not a rational chain, $M$ is prime, and not a lens space (see \\cite[6.1]{Neumann-plumbing_graphs} for lens space graphs), so by \\cite[2.1.2]{AFW_3M} $\\pi_{1}(M)\\not\\cong \\pi_{1}(M')$. The result follows because $\\pi_{1}^{\\infty}(S_{p_{1},p_{2}})=\\pi_{1}(M)$ \\cite{Mumford-surface_singularities}.\n\\end{proof}\n\n\n\n\\@startsection{subsection}{3{Almost minimal models}\\label{sec:peeling}\n\nWe now recall the construction of an almost minimal model for a log smooth pair $(X,D)$. We consider only the case when $X\\setminus D$ is affine, for a complete treatment we refer to \\cite[\\S 3]{Miyan-OpenSurf}, see p.\\ 107 loc.\\ cit.\\ for a summary. The exposition here is based on \\cite[\\S 3]{Palka-minimal_models}.\n\nThe motivation for this construction to get an explicit description of a log MMP run for the pair $(X,D)$, such that the log terminal singularities on the image of $(X,D)$ are introduced as late as possible: in fact, \\emph{an almost minimal model} is a log resolution of the minimal model in the sense of Mori. Having understood the latter, one can use this theory to reconstruct the initial pair $(X,D)$. \n\nThe almost minimal model will be created by iterating birational morphisms of the following type\n\n\\begin{dfn}[{half-point attachment, cf.\\ \\cite{Fujita-noncomplete_surfaces}}]\\label{def:hp}\nLet $(X,D)$ be a log smooth pair. Assume that\n\\begin{equation}\\label{eq:hp}\nA\\subseteq X\\mbox{ is a }(-1)\\mbox{-curve such that } A\\not\\subseteq D\\mbox{ and } A\\cdot D=1.\n\\end{equation}\nThe composition of the contraction of $A$ and new superfluous $(-1)$-curves in the subsequent images of $D$:\n\\begin{equation*}\n\\phi_{A}\\colon (X,D)\\to (X',D'),\n\\end{equation*} \nwhere $D'=(\\phi_{A})_{*}D$, is called a \\emph{half-point attachment} (of $A$).\n\\end{dfn}\n\\begin{rem*}\n\tIn general, $\\phi_{A}$ is \\emph{not} uniquely determined by $A$. Indeed, it may happen that after some contraction within $\\phi_{A}$, there appear two superfluous $(-1)$-curves. For example, if $A$ meets the middle component $T'$ of a subchain $T=[2,2,2]$ of $D$, whose components are non-branching in $D$, then after the contraction of $A+T'$, both components of the image of $T$ are superfluous $(-1)$-curves in the image of $D$, but only one of them can be contracted.\n\t\n\tHowever, such ambiguity never happens if $\\mathbb{C}[X\\setminus D]$ is a UFD. Indeed, Lemma \\ref{lem:producing_Cst}\\ref{item:HP_C} shows that in this case $\\phi_{A}$ contracts exactly $A$ and a maximal $(-2)$-twig of $D$ meeting $A$ (if such occurs).\n\\end{rem*}\n\n\n\\begin{lem}\\label{lem:HPD_k}\n\tLet $(X,D)\\to (X',D')$ be a half-point attachment. Then for all $m\\geq 0$ we have $h^{0}(m(K_{X'}+D'))=h^{0}(m(K_{X}+D))$. In particular, $\\kappa(X'\\setminus D')=\\kappa(X\\setminus D)$.\n\\end{lem}\n\\begin{proof}\n\tLet $A$ be as in \\eqref{eq:hp}. We have $A\\cdot(K_{X}+D+A)=-1<0$, so for any $m>0$, $mA$ is in the fixed locus of $|m(K_{X}+D+A)|$. Therefore, $|m(K_{X}+D)| \\cong |m(K_{X}+D+A)| \\cong |m(K_{X'}+D')|$ because $X\\to X'$ contracts only components of $D+A$.\n\\end{proof}\n\n\\begin{cor}\\label{cor:S_kappa}\n\tThe surfaces $S_{p_1,p_2}$ from Theorem \\ref{THM} are in $\\mathcal{S}_{0}$.\n\\end{cor}\n\\begin{proof}\n\tBy Corollary \\ref{cor:S_k}, $S_{p_1,p_2}\\in \\mathcal{S}_{\\kappa}$ for some $\\kappa$. Construction \\ref{constr} provides a log smooth completion of $S_{p_1,p_2}$ obtained from a log smooth completion of $\\mathbb{C}^{*}\\times \\mathbb{C}^{*}$ by two half-point attachments. Hence by Lemma \\ref{lem:HPD_k}, $\\kappa=\\kappa(S_{p_1,p_2})=\\kappa(\\mathbb{C}^{*}\\times \\mathbb{C}^{*})=0$.\n\\end{proof}\n\nWe now state the main result of the theory of peeling. Recall that $D^{\\#}=D-\\operatorname{Bk} D$, and $\\operatorname{Bk} D$ is a $\\mathbb{Q}$-effective divisor with coefficients in $(0,1)$, supported on the twigs of $D$.\n\n\\begin{prop}[Construction of an almost minimal model]\\label{prop:MMP}\n\tLet $(X,D)$ be a minimal log smooth completion of a smooth affine surface. Then there is a birational morphism\n\t\t\\begin{equation}\\label{eq:MMP}\n\t\t\\psi\\colon (X,D)\\toin{\\psi_{1}}(X_{1},D_{1})\\toin{\\psi_{2}}\\dots \\toin{\\psi_{n}} (X_{n},D_{n})\n\t\t\\end{equation}\n\tsuch that, for all $i\\in \\{1,\\dots, n\\}$:\n\t\\begin{enumerate}\n\t\t\\item\\label{item:psi_hpd} $\\psi_{i}$ is a half-point attachment.\n\t\t\\item\\label{item:X_i-smooth} $D_{i}=(\\psi_{i})_{*}D_{i-1}$ is snc-minimal, $(X_{i},D_{i})$ is log smooth and $X_{i}\\setminus D_{i}$ is an affine open subset of $X\\setminus D$.\n\t\t\\item\\label{item:psi_kappa} $h^{0}(m(K_{X_{i}}+D_{i}))=h^{0}(m(K_{X}+D))$ for all $m\\geq 0$. In particular, $\\kappa(X_{i}\\setminus D_{i})=\\kappa(X\\setminus D)$.\n\t\t\\item\\label{item:psi_min} If $\\kappa(X\\setminus D)\\geq 0$ then $K_{X_{n}}+D_{n}^{\\#}$ is nef. Otherwise, either $X_{n}$ admits a $\\P^{1}$-fibration with $(K_{X_{n}}+D_n^{\\#})\\cdot F<0$ for a fiber $F$; or $-(K_{X_{n}}+D_{n})$ becomes ample after the contraction of $\\operatorname{Bk} D_n$.\n\t\\end{enumerate}\n\\end{prop}\n\\begin{proof}[Sketch of a proof]\n\tLet $\\alpha\\colon (X,D)\\to (Y,D_{Y})$ be the contraction of $\\operatorname{Supp}\\operatorname{Bk} D$, i.e.\\ of all admissible twigs of $T$. Then $\\alpha^{*}(K_{Y}+D_{Y})=K_{X}+D^{\\#}$. Indeed, for every component $T_{0}$ of $\\operatorname{Bk} D$, we have $T_{0}\\cdot (K_{X}+D-\\alpha^{*}(K_{Y}+D_{Y}))=T_{0}\\cdot(K_{X}+T_{0})+\\beta_{D}(T_{0})=-2+\\beta_{D}(T_{0})=T_{0}\\cdot \\operatorname{Bk} D$ by definition of bark, so $K_{X}+D-\\alpha^{*}(K_{Y}+D_{Y})=\\operatorname{Bk} D$ because $\\operatorname{Bk} D$ is negative definite.\n\t\n\tIn particular, $(Y,D_{Y})$ is log terminal. General theorems of Mori theory imply that either $(Y,D_{Y})$ is minimal, so \\ref{item:psi_min} holds for $n=0$, or there is a log extremal curve $\\ll\\subseteq Y$ such that $\\ll^{2}<0$ and $\\ll\\cdot (K_{Y}+D_{Y})<0$. Put $A=\\alpha^{-1}_{*}\\ll$. We claim that $A$ satisfies \\eqref{eq:hp}.\n\t\n\tWe have $0>\\ll\\cdot (K_{Y}+D_{Y})=A \\cdot(K_{X}+D^{\\#})$. Suppose $A\\subseteq D$. Then $0>-2+\\beta_{D}(A)-A\\cdot \\operatorname{Bk} D$, hence $A$ is a tip of $D-\\operatorname{Supp}\\operatorname{Bk} D$ and meets at most one component of $\\operatorname{Bk} D$. This means that $A$ is a component of an admissible twig of $D$, a contradiction with the definition of $\\operatorname{Bk} D$.\n\t\n\tTherefore, $A\\not \\subseteq D$, so $A\\cdot D^{\\#}>0$. Thus $A\\cdot K_{X}<0$, which together with $A^{2}<0$ implies that $A$ is a $(-1)$-curve. Moreover, $A\\cdot D^{\\#}<-A\\cdot K_{X}=1$, so $A$ meets $D$ only on $\\operatorname{Bk} D$. Now a computation \\cite[II.3.7.1(2)]{Miyan-OpenSurf}, which is rather complicated, but uses nothing more than negative definiteness of $A+\\operatorname{Bk} D$, shows that $A$ meets each connected component of $D$ at most once. Because $X\\setminus D$ is affine, it follows that $A\\cdot D=1$, so $A$ satisfies \\eqref{eq:hp}, as claimed.\n\t\n\tWe now put $\\psi_{1}=\\phi_{A}\\colon (X,D)\\to(X_{1},D_{1})$, for some choice of $\\phi_{A}$ such that $D_{1}$ is snc-minimal. Then for $i=1$, \\ref{item:psi_hpd},\\ref{item:X_i-smooth} hold by definition and \\ref{item:psi_kappa} holds by Lemma \\ref{lem:HPD_k}. Next, we replace $(X,D)$ by $(X_{1},D_{1})$ and repeat the procedure, which eventually ends since $\\rho(X)$ drops after each step.\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{THM}}\\label{sec:proof}\n\\setcounter{claim}{0}\n\nRecall that $S_{p_1,p_2}\\in \\mathcal{S}_{0}$ by Corollary \\ref{cor:S_kappa}, and $S_{p_1,p_2}\\not\\cong S_{p_1',p_2'}$ for $\\{p_1,p_2\\}\\neq \\{p_1',p_2'\\}$ by Corollary \\ref{cor:st}\\ref{item:distinct}. It remains to show that any $S\\in \\mathcal{S}_{0}$ is isomorphic to some $S_{p_1,p_2}$. Let $(X,D)$ be a minimal log smooth completion of $S$. The idea of the proof is to view $\\phi$ from Construction \\ref{constr} as some almost minimalization \\eqref{eq:MMP} of $(X,D)$. By Proposition \\ref{prop:MMP} each intermediate pair $(X_{i},D_{i})$ in \\eqref{eq:MMP} is a log smooth completion of an affine open subset of $X\\setminus D$, of the same Kodaira dimension. Hence under the assumptions of Theorem \\ref{THM}, $\\kappa(X_{i}\\setminus D_{i})=0$ and $\\mathbb{C}[X_{i}\\setminus D_{i}]$ is a UFD by Remark \\ref{rem:open_UFD}. \n\nTo reconstruct $(X,D)$ we need to understand each half-point attachment $\\psi_{i+1}\\colon (X_{i},D_{i})\\to(X_{i+1},D_{i+1})$. This is done in Lemma \\ref{lem:producing_Cst}, which is a version of \\cite[Lemma III.4.4.3]{Miyan-OpenSurf}, with almost the same proof.\n\n\\begin{lem}\\label{lem:producing_Cst}\n\tLet $S$ be a smooth affine surface. Assume that $\\kappa(S)=0$ and $\\mathbb{C}[S]$ is UFD. Let $(X,D)$ be a minimal log smooth completion of $S$ and let $\\phi_{A}$ be a half-point attachment of some $A\\subseteq X$. Then\n\t\\begin{enumerate}\n\t\t\\item\\label{item:HP_Cst} $A\\cap S\\cong \\mathbb{C}^{1}$ is a smooth fiber of some $\\mathbb{C}^{*}$-fibration of $S$\n\t\t\\item\\label{item:HP_C} $\\operatorname{Exc} \\phi_{A}=[1,(2)_{k}]$, for some $k\\geq 0$, is a twig of $D+A$. Its image does not lie on a rational twig of $(\\phi_A)_{*}D$.\n\t\\end{enumerate}\n\t\\begin{figure}[htbp]\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\begin{subfigure}[t]{0.4\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.34]{figures\/HP}\n\t\t\t\\end{subfigure}\n\t\t\t&\n\t\t\t\\raisebox{1cm}{$\\xrightarrow{\\quad \\phi_{A}\\quad}$}\n\t\t\t&\n\t\t\t\\begin{subfigure}[t]{0.2\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.34]{figures\/HP_1}\n\t\t\t\\end{subfigure}\n\t\t\\end{tabular}\n\t\t\\caption{A half-point attachment to $S$ if $\\mathbb{C}[S]$ is a UFD and $\\kappa(S)=0$, see Lemma \\ref{lem:producing_Cst}.}\n\t\t\\label{fig:HP}\n\t\\end{figure}\n\\end{lem}\n\\begin{proof}\n\\ref{item:HP_Cst} Because $\\mathbb{C}[S]$ is a UFD, $A|_{S}$ is a divisor of zeros of some $f\\in \\mathbb{C}[S]$. Let $F_{\\mathrm{aff}}$ be some general fiber of $f$. It is reduced and irreducible because $A|_{S}$ is. Put $S'=S\\setminus A$, $U'=f(S')\\subseteq \\mathbb{C}^{*}$. By Lemma \\ref{lem:HPD_k} $\\kappa(S')=\\kappa(S)=0$, so the Kawamata addition theorem \\cite[Theorem 11.15]{Iitaka_AG} gives $0=\\kappa(S')\\geq \\kappa(F_{\\mathrm{aff}})+\\kappa(U')\\geq \\kappa(F_{\\mathrm{aff}})+\\kappa(\\mathbb{C}^{*})=\\kappa(F_{\\mathrm{aff}})\\geq 0$, where the last inequality follows from the Iitaka Easy Addition theorem. Hence $\\kappa(F_{\\mathrm{aff}})=0$, which implies that $\\kappa(F_{\\mathrm{aff}})\\cong \\mathbb{C}^{*}$ because $F_{\\mathrm{aff}}$ is smooth and affine. Thus $f$ is a $\\mathbb{C}^{*}$-fibration of $S$, with $A|_{S}$ as one of its fibers.\n\n\\ref{item:HP_C} Let $T$ be a maximal admissible twig of $D$ meeting $A$ in its first tip; put $T=0$ if there is no such. Since $S$ is affine, $D\\neq T$. Let $C$ be the component of $D-T$ meeting $A+T$.\nLet $\\tau\\colon \\hat{X}\\to X$ be a minimal sequence of blowups over $D$ such that the $\\mathbb{C}^{*}$-fibration from \\ref{item:HP_Cst}  extends to a $\\P^{1}$-fibration of $\\hat{X}$; let $F$ be a fiber containing $\\hat{A}:= \\tau^{-1}_{*}A$. Put $\\hat{D}=(\\tau^{*}D)_{\\mathrm{red}}$. Note that $\\tau$ does not touch $\\hat{A}$: indeed, otherwise the only base point of $\\tau^{-1}$ is the unique common point of $A$ and $D$, so  $F\\subseteq \\tau^{*}A$ is negative definite, which is false.\n\nIf a $(-1)$-curve $L\\subseteq \\hat{D}\\vert$ is superfluous in $\\hat{D}$, then $L$ is not a component of a rational twig of $\\hat{D}$. Indeed, $L\\not\\subseteq \\operatorname{Exc}\\tau$ since $\\tau$ is minimal, and $\\tau(L)^{2}\\neq -1$ since $D$ is snc-minimal; hence $\\tau(L)^{2}\\geq 0$, and the claim follows from Lemma \\ref{lem:rational-twigs-are-admissible}. Let $\\sigma$ be the contraction of all vertical superfluous $(-1)$-curves in $\\hat{D}$ and its images. By our claim, $\\sigma$ does not touch rational twigs of $\\hat{D}$. Put $D'=\\sigma_{*}\\hat{D}$, $F'=\\sigma_{*}F$, $A':= \\sigma_{*}\\hat{A}$.\n\nIf $\\sigma$ touches $\\hat{A}$ then $T=0$ and $\\tau^{-1}_{*}C\\subseteq \\operatorname{Exc}\\sigma$ does not lie in a rational twig of $\\hat D$, so the result follows. Assume $\\sigma$ does not touch $\\hat A$, so $A'$ is a $(-1)$-curve of multiplicity one in a fiber $F'$. Now,  $F'-A'$ contains another $(-1)$-curve, say $C'$. By definition of $\\sigma$, $\\beta_{D'}(C')\\geq 3$. Since $\\beta_{F'_{\\mathrm{red}}}(C')\\leq 2$, $C'$ meets $D'_{\\mathrm{hor}}$. By Lemma \\ref{lem:untwisted}, $D'_{\\mathrm{hor}}$ consists of two $1$-sections, so $C'$ has multiplicity one in $F'$. Hence $\\beta_{F'_{\\mathrm{red}}}(C')=1$, so $C'\\cdot D'_{\\mathrm{hor}}=2$, which implies that $(F'-C')\\cdot D'_{\\mathrm{hor}}=0$, and that $C'$ is a unique $(-1)$-curve in $F'-A'$. Therefore, $F'=[1,(2)_{k},1]$ for some $k\\geq 0$. In other words, $T':= F'-A'-C'=[(2)_{k}]$ is zero or a maximal twig of $D'$.  Applying Lemma \\ref{lem:rational-twigs-are-admissible} to image of $D'$ after contraction of $A'+T'$, we infer that $C'$ does not lie in a rational twig of $D'-T'$. \n\nSince $\\sigma$ does not touch rational twigs of $D'$, $\\hat{T}:= \\sigma^{-1}_{*}T'$ is zero or a maximal twig of $\\hat{D}$; and the component $\\hat{C}:= \\sigma^{-1}_{*}C'$ meeting it does not lie in a rational twig of $\\hat{D}-\\hat{T}$. Thus $\\tau_{*}\\hat{T}=T$, $\\tau_{*}\\hat{C}=C$: indeed, otherwise $\\tau$ touches $\\hat{C}$, so $C^{2}\\geq 0$, and $C$ lies in an admissible twig of $D$, which is impossible by Lemma \\ref{lem:rational-twigs-are-admissible}. Therefore, $T=[(2)_{k}]$, and $C$ does not lie in any rational twig of $D$, as claimed.\n\\end{proof}\n\n\nOur next lemma is a standard result concerning curves of canonical type, see eg.\\ \\cite[I.3.3.1]{Miyan-OpenSurf}. It will give a description of the almost minimal model of $(X,D)$, or, more precisely, the pair obtained from $(X,D)$ after all possible half point attachments, called a \\emph{strongly minimal model} in \\cite[II.4.9]{Miyan-OpenSurf}.\n\nRecall that $\\mathbb{F}_{m}$ for an integer $m\\geq 0$ denotes the $m$-th Hirzebruch surface $\\P(\\O_{\\P^{1}}\\oplus \\O_{\\P^{1}}(m))$.\n\n\n\\begin{lem}[strongly minimal models]\\label{lem:O}\n\tLet $S$ be a smooth rational affine surface such that $\\mathbb{C}[S]$ is a UFD. Let $(X,D)$ be a minimal log smooth completion of $S$. Assume that $K_{X}+D\\equiv 0$ and that all $(-1)$-curves on $X$ are contained in $D$. Then  one of the following holds:\n\t\t\\begin{enumerate}\n\t\t\t\\item\\label{item:Fm_4} $X=\\mathbb{F}_{m}$ for some $m\\neq 1$ and  $D=\\lc0,m,0,-m)\\!)$,\n\t\t\t\\item\\label{item:Fm_3} $X=\\mathbb{F}_{m}$ for some $m\\neq 1$ and  $D=\\lc0,m,-m-2 )\\!)$,\n\t\t\t\\item\\label{item:triangle} $X=\\P^{2}$ and  $D=(\\!( -1,-1,-1 )\\!)$, i.e.\\ $D$ is a triangle,\n\t\t\t\\item\\label{item:conic} $X=\\P^{2}$ and  $D=(\\!( -1,-4)\\!) $, i.e.\\ $D$ is a sum of a line and a conic which meet at two points.\n\t\t\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\tLet $\\tau\\colon (X,D)\\to (X',D')$ be a contraction of some $(-1)$-curves in $D$ and its images, such that all singularities of $D'$ are ordinary double points. Since $D$ is snc-minimal, $\\operatorname{Bs}\\tau^{-1}\\subseteq \\operatorname{Sing} D$. Let $A'$ be a $(-1)$-curve in $X'$. Then $A'\\cdot D'=-A'\\cdot K_{X'}=1$. If $A'\\not\\subseteq D'$ then $A'\\cap \\operatorname{Sing} D'=\\emptyset$, so $\\tau$ does not touch $A:=\\tau^{-1}_{*}A$. Therefore, $A\\subseteq X$ is a $(-1)$-curve not contained in $D$, contrary to our assumption. Hence $A'\\subseteq D'$, so $\\beta_{D'}(A')=A'\\cdot D'-A^2=2$. Since no two components of $D'$ are tangent, $\\#A'\\cap (D'-A')=2$. After the contraction of $A'$, the singularities of the image of $D'$ are still at most ordinary double points. \n\t\n\tThus we can assume that $X'$ has no $(-1)$-curves, i.e.\\ $X'=\\P^2$ or $\\mathbb{F}_{m}$ for some $m\\neq 1$. By Lemma \\ref{lem:S0}\\ref{item:UFD}, the components of $D$ generate $\\operatorname{Pic}(X)$, so the components of $D'$ generate $\\operatorname{Pic}(X')$, too. \n\t\n\tAssume first that $X'=\\P^{2}$. Then $D'\\equiv -K_{\\P^{2}} \\equiv 3\\cdot [\\mbox{line}]$. Because the components of $D'$ generate $\\operatorname{Pic}(\\P^{2})$, $D'$ is reducible. In particular, $D'$ has no singular component, so $\\tau=\\mathrm{id}$ and $D$ is as in \\ref{item:triangle} or \\ref{item:conic}.\n\t\n\tAssume now that $X'=\\mathbb{F}_{m}$. Then $2=\\rho(X')\\leq \\#D'$. Because $X'\\setminus D'$ is affine, $D'$ is connected, so $\\beta_{D'}(C)\\geq 1$ for every component $C$ of $D'$. Hence $0=C\\cdot (K_{X'}+D')=2p_{a}(C)-2+\\beta_{D'}(C)\\geq 2p_{a}(C)-1$, It follows that $C$ is smooth, so $\\tau=\\mathrm{id}$. Moreover, $C$ is rational and $\\beta_{D}(C)=2$, so $D$ is circular. \n\t\n\tDenote by $F,T\\in\\operatorname{Pic}(\\mathbb{F}_{m})$ the classes of the fiber of $\\mathbb{F}_{m}$ and of the section with $T^{2}=-m$, which is unique if $m\\neq 0$. We have $F\\cdot D=-F\\cdot K_{\\mathbb{F}_{m}}=2$, so by Lemma \\ref{lem:untwisted} $D_{\\mathrm{hor}}$ consists of two $1$-sections, say $H_{i}\\equiv a_{i}F+T$ for $i\\in\\{1,2\\}$. We can assume $a_1\\geq a_2$. Note that either $a_{i}=0$ or $0\\leq H_{i}\\cdot T=a_{i}-m$, so $a_{i}\\geq m$. Hence $a_1\\geq m$ and either $a_2\\geq m$ or $H_2\\equiv T$. Let $v$ be the number of fibers in $D$. Because all fibers are disjoint and meet $H_{1}$ and $H_{2}$, from the fact that $\\beta_{D}(H_j)=2$, $j\\in \\{1,2\\}$ we infer that \n\t\\begin{equation}\\label{eq:Hirz}\n\t2-v=H_{1}\\cdot H_{2}=a_{1}+a_{2}-m.\n\t\\end{equation}\n\tSuppose $v=0$. Then because $H_{1},H_{2}$ generate $\\operatorname{Pic}(\\mathbb{F}_{m})=\\mathbb{Z}[F+T]$, we have $\\det \\left[ \\begin{smallmatrix} a_{1} & 1 \\\\ a_{2} & 1 \\end{smallmatrix}\\right]=\\pm 1$, so $a_{1}-a_{2}=1$. Now \\eqref{eq:Hirz} gives $a_{2}=\\tfrac{1}{2}(m+1)$, so $2\\nmid m$, hence $m\\geq 3$ and $0\\neq a_2<m$, a contradiction\n\t\n\tThus $v\\in \\{1,2\\}$. It follows that $H_{2}\\equiv T$, because otherwise $a_{1},a_{2}\\geq m$ and \\eqref{eq:Hirz} gives $1\\geq  a_{1}+a_{2}-m\\geq m$, so $m=0$ and $1\\geq a_{1}+a_{2}\\geq 2$, which is false. Hence $a_{2}=0$, $H_{2}^{2}=-m$ and by \\eqref{eq:Hirz} $a_{1}=m+2-v$, so $H_{1}^{2}=m+4-2v$. We conclude that \\ref{item:Fm_4} holds if $v=2$ and \\ref{item:Fm_3} holds if $v=1$.\n\\end{proof}\n\n\\begin{rem*}\n  Lemma \\ref{lem:O} will imply that almost minimal models for surfaces in $\\mathcal{S}_{0}$ are of type $(O)$ in \\cite[8.70]{Fujita-noncomplete_surfaces}, cf.\\ \\cite[Proposition 1.5(1)]{Kojima_k0}. For definition and basic properties of this type see \\cite[8.8, 8.12]{Fujita-noncomplete_surfaces}. Parts \\ref{item:Fm_4}-\\ref{item:conic} correspond to cases 4, 6, 8 of 9 in \\cite[Theorem 3.1]{Kojima_k0}. We will not use these consequences.\t\n\\end{rem*}\n\n\n\\begin{proof}[Proof of Theorem \\ref{THM}]\n\tLet $(X,D)$ be a minimal log smooth completion of $S\\in \\mathcal{S}_{0}$ and let \n\t\\begin{equation*}\n\t\\psi\\colon (X,D)\\to (X_{n},D_{n})\n\t\\end{equation*}\n\tbe an almost minimalization as in \\eqref{eq:MMP}. By Proposition \\ref{prop:MMP}\\ref{item:psi_min} $\\kappa(X_n\\setminus D_n)=\\kappa(X\\setminus D)=0$, so \\cite[II.6.2.1]{Miyan-OpenSurf} implies that $K_{X_{n}}+D_{n}^{\\#}\\equiv 0$, that is, $K_{X_{n}}+D_{n}\\equiv \\operatorname{Bk} D_{n}$. By Proposition \\ref{prop:MMP}\\ref{item:psi_hpd} $\\psi$ is a sequence of half-point attachments, so by Lemma \\ref{lem:producing_Cst}\\ref{item:HP_C} no base point of $\\psi^{-1}$ lies on an admissible twig of $D_n$, i.e.\\ $\\operatorname{Bs}\\psi^{-1}\\cap \\operatorname{Supp}\\operatorname{Bk} D_{n}=\\emptyset$. Hence $\\psi^{-1}_{*}\\operatorname{Bk} D_{n}=\\psi^{*}\\operatorname{Bk} D_{n}\\equiv \\psi^{*}(K_{X_{n}}+D_{n})$, which by Lemma \\ref{lem:S0} writes uniquely as a sum of components of $D$, with integral coefficients. But $\\psi^{-1}_{*}\\operatorname{Bk} D_{n}$ is a sum of components of $D$ with coefficients in $[0,1)$. Hence $\\psi^{-1}_{*}\\operatorname{Bk} D_{n}=0$, which implies that\n\t\\begin{equation}\\label{eq:K+D=0}\n\tK_{X_{n}}+D_{n}\\equiv 0.\n\t\\end{equation}\n\tWe claim that there is a sequence of half-point attachments\n\t\\begin{equation}\\label{eq:claim}\n\t\\phi\\colon (X,D)\\to (Z,D_{Z}) \\mbox{ such that } (Z,D_Z) \\mbox{ is as in Lemma \\ref{lem:O}}.\n\t\\end{equation}\n\tIndeed, let $(X_{n},D_{n})\\to (Z',D_{Z'})$ be any sequence of half-point attachments. By Remark \\ref{rem:open_UFD},  $\\mathbb{C}[Z'\\setminus D_{Z}']$ is a UFD, and $K_{Z'}+D_{Z'}\\equiv 0$  by \\eqref{eq:K+D=0}. Let $A\\subseteq Z'$ be a $(-1)$-curve, $A\\not\\subseteq D_{Z'}$.  Then $A\\cdot D_{Z'}=-A\\cdot K_{Z'}=1$, so $A$ satisfies \\eqref{eq:hp}. After a half-point attachment of $A$, the image of $Z'$ has smaller rank, so after finitely many half-point attachments, we get $(Z,D_{Z})$ with no $(-1)$-curves off $D_Z$, as needed.\n\t\n\tWe need to show that, for a suitable log smooth completion $(X,D)$ of $S$, the morphism $\\phi$ from \\eqref{eq:claim} can be chosen as in Construction \\ref{constr}. Denote by $A_{1},\\dots, A_{r}$ the curves not contained in $D$ which are contracted by $\\phi$ (in particular, $r \\geq n$). Write $\\phi(A_{i})=\\{b_{i}\\}\\subseteq Z$ for $i\\in \\{1,\\dots, r\\}$. By Lemma \\ref{lem:producing_Cst}\\ref{item:HP_C} each $b_{i}$ lies on exactly one component $C_{i}$ of $D_{Z}$. Lemma \\ref{lem:S0} gives $\\#D=\\rho(X)$, so\n\t\\begin{equation*}\n\t\\#D_{Z}-\\rho(Z)=\\#D+r-\\rho(X)=r,\n\t\\end{equation*}\n\thence $r=2$ if $(Z,D_Z)$ is as in Lemma \\ref{lem:O}\\ref{item:Fm_4},\\ref{item:triangle} and $r=1$ if $(Z,D_Z)$ is as in Lemma \\ref{lem:O}\\ref{item:Fm_3},\\ref{item:conic}. \n\t\n\tLet $R_{1},\\dots, R_{r}$ be a basis of $\\ker[\\mathbb{Z}[D_{Z}]\\to \\operatorname{Pic}(Z)]$, and let $a_{ij}$ be the coefficient of $C_{i}$ in $R_{j}$. We have $\\operatorname{Pic}(X)=\\operatorname{Pic}(Z)\\oplus\\mathbb{Z}[\\operatorname{Exc}\\phi]=\\mathbb{Z}[\\phi^{*}D_{Z}]\/(\\phi^{*}R_{1},\\dots,\\phi^{*}R_{i})$. Because by Lemma \\ref{lem:producing_Cst}\\ref{item:HP_C} each $A_{i}$ has multiplicity $1$ in $\\phi^{*}C_{i}$ and $0$ in $\\phi^{*}(D_{Z}-C_{i})$, Lemma \\ref{lem:S0}\\ref{item:units} gives \n\t\\begin{equation}\\label{eq:det}\n\t\\det [a_{ij}]_{1\\leq i,j\\leq r}=\\pm 1.\n\t\\end{equation}\n\n\t\n\tNow we treat each of the cases \\ref{item:Fm_4}--\\ref{item:conic} from Lemma \\ref{lem:O} separately. We would like to reduce everything to case \\ref{item:Fm_4} with $m=0$. So in each of the remaining cases, we  construct $\\sigma\\colon Z\\dashrightarrow Z'$, which is an isomorphism on $Z\\setminus D_{Z}$ and on some neighborhood of $b_{1},\\dots, b_{r}$; and $\\phi''\\colon (Z',\\sigma_{*}D_{Z})\\to (Z'',D_{Z''})$, which is an identity or a half-point attachment, such that $(D_{Z''},D_{Z''})$ is as in one of the cases not considered yet. Then we lift $\\sigma$ so that the following diagram is commutative:\n\t\\begin{equation*}\n\t\\xymatrix{\n\t\t(X,D) \\ar[r]^{\\phi} \\ar@{-->}[d]^{\\sigma'} & (Z,D_{Z}) \\ar@{-->}[d]^{\\sigma} & \\\\\t\n\t\t(X',D') \\ar[r]^{\\phi'} & (Z',D_{Z'}) \\ar[r]^{\\phi''} & (Z'',D_{Z''})\n\t}\n\t\\end{equation*}\n\twhere $D'$, $D_{Z'}$, $D_{Z''}$ are reduced total transforms of $D$, and $(X',D')$ is another log smooth completion of $S$. Eventually, we replace $\\phi$ with $\\phi''\\circ \\phi'$, which is another sequence of  half-point attachments. \n\t\n\tConsider case \\ref{item:conic}, i.e.\\ $r=1$, $D_{Z}$ is a sum of a line $L$ and a conic $C$, and $\\#L\\cap C=2$. We have $R_{1}=C-2L$, so by \\eqref{eq:det} $b_{1}\\in C$. Define $\\sigma$ as a blow up at a point of $C\\cap L$ followed by a flow (see Section \\ref{sec:standard}) on the proper transform of $L$. Then $(Z',D_{Z}')$ is as in case \\ref{item:Fm_3}, so we put $\\phi''=\\mathrm{id}$.\n\t\n\tConsider case \\ref{item:triangle} and let $L\\subseteq D_{Z}$ be the line not containing $b_{1},b_{2}$. As before, we define $\\sigma$ as a blowup at $L\\cap (D_{Z}-L)$, followed by a flow on the proper transform of $L$. Then $(Z',D_{Z}')$ is as in case \\ref{item:Fm_4}.\n\t\n\tConsider case \\ref{item:Fm_3}. Write $D_{Z}=F+T+H$, where $F^{2}=0$, $T^{2}=-m$ and $H^{2}=m+2$. We can assume that $b_{1}\\in F$: indeed, otherwise after $\\pm C_{1}^{2}$ flows on $F$ we obtain a pair with that property. Now $r=1$ and $R_{1}=(m+1)F+T-H$, so \\eqref{eq:det} gives $m=0$. Let $\\sigma$ be a flow on $T$ such that $(\\sigma_{*}F)^{2}=1$, followed by a blowup at $\\sigma(F\\cap H)$. Let now $\\phi''$ be the contraction of a proper transform of a member of $|T|$ passing through $F\\cap H$. The resulting pair is as in \\ref{item:Fm_4}.\n\t\n\tEventually, consider case \\ref{item:Fm_4}. Write $D_{Z}=F_{1}+T+F_{2}+H$, where $F_{1}^{2}=F_{2}^{2}=F_{1}\\cdot F_{2}=0$ and as before $T^{2}=-m$, $b_{1},b_{2}\\not\\in T$. We have $r=2$ and $R_{1}=F_{1}-F_{2}$, $R_{2}=mF_{2}+T-H$, so \\eqref{eq:det} implies (after renaming $b_{i}$, $F_{i}$ if necessary) that $b_{1}\\in F_{1}$, $b_{2}\\in H+T$. After $m$ flows on $F_{2}$ we can assume that $m=0$, so $Z\\cong \\P^{1}\\times \\P^{1}$. We can choose coordinates on $Z$ such that $F_{1}=\\ll_{1,0}$, $F_{2}=\\ll_{1,\\infty}$, $b_2\\in H=\\ll_{2,0}$, $T=\\ll_{2,\\infty}$, $b_{1}=(0,1)$, $b_{2}=(1,0)$. Now by Lemma \\ref{lem:producing_Cst}\\ref{item:HP_C}, $\\phi\\colon (X,D)\\to (Z,D_{Z})$ is as in Construction \\ref{constr}.\n\\end{proof}\n\n\n\\section{Diffeomorphism type of $S_{p_{1},p_{2}}$}\\label{sec:diff}\n\nIn this section we prove Theorem \\ref{thm:diff}. To this end, we need to translate Construction \\ref{constr} to the language of Kirby calculus, where blowing up corresponds to attaching handles. We give a quick overview of handlebodies and Kirby diagrams in Section \\ref{sec:Kirby_overview}.  In Proposition  \\ref{prop:handles} we show that Construction \\ref{constr} amounts to attaching two $2$-handles to $\\mathbb{T}^{2}\\times \\mathbb{D}^{2}$, whose interior is identified with $\\mathbb{C}^{*}\\times \\mathbb{C}^{*}$, see Figure \\ref{fig:handles}. Theorem \\ref{thm:diff} follows by sliding its $2$-handle over the attached ones. The boundary $M$ of the obtained $4$-manifold is a $0$-surgery on the knot $K$ from Figure \\ref{fig:knot}. It is interesting to note that this is one of the few \\emph{exceptional} surgeries on $K$, classified in  \\cite{BW_surgery}. More precisely, while all but finitely many surgeries on $K$ are hyperbolic, $M$ contains incompressible tori. Decomposition of $M$ along these tori is described in Proposition \\ref{prop:JSJ}. While interesting in its own right, it gives another way to distinguish $S_{p_1,p_2}$, for different $\\{\\deg p_{1}, \\deg p_{2}\\}$. \n\\smallskip\n\nWe begin with the following observation.\n\\begin{prop}\\label{prop:htp}\n\tFor every $p_{1},p_{2}$, $S_{p_1,p_2}$ is homotopically equivalent to $\\mathbb{S}^{2}$. \n\\end{prop}\n\\begin{proof}\t\n\tViewing $S=S_{p_1,p_2}$ as an iterated affine modification of $\\mathbb{C}^{2}$, we infer from \\cite[Lemma 3.4]{Kal_exotic-measures} that $\\pi_{1}(S)=\\pi_{1}(\\mathbb{C}^{2})=\\{1\\}$. This modification replaces $\\{x_1x_2=0\\}\\subseteq \\mathbb{C}^{2}$, which is contractible, by the affine part of $A_1+A_2$, which is a disjoint union of two copies of $\\mathbb{C}^{1}$. Hence $e_{\\mathrm{top}}(S)=e_{\\mathrm{top}}(\\mathbb{C}^{2})-1+2=2$. Because $S$, being Stein, has a homotopy type of a CW-complex of (real) dimension two \\cite[11.2.6]{GS_Kirby}, it follows that $H_{i}(S,\\mathbb{Z})=\\mathbb{Z}$ for $i=2$ and $0$ for $i>2$. Thus $S$ has weak homotopy type of $\\mathbb{S}^{2}$. Applying the Whitehead theorem to the continuous map $\\mathbb{S}^{2}\\to S$ given by a generator of $\\pi_{2}(S)=H_{2}(S;\\mathbb{Z})=\\mathbb{Z}$, we infer that $\\mathbb{S}^{2}\\simeq_{\\mathrm{htp}} S$, as claimed.\n\\end{proof}\n\n\\@startsection{subsection}{3{Overview of Kirby calculus}\\label{sec:Kirby_overview}\n\nWe now briefly recall the language of Kirby diagrams of $4$-manifolds, for a complete treatment see \\cite[\\S 4-5]{GS_Kirby}. A ($4$-dimensional) $k$-handle $h$ for $k\\in \\{0,\\dots, 4\\}$ is a copy of $\\mathbb{D}^{k}\\times \\mathbb{D}^{4-k}$ attached to a boundary of a $4$-manifold $V$ via an embedding $\\phi\\colon \\d\\mathbb{D}^{k}\\times \\mathbb{D}^{4-k}\\to \\d V$. The images of $\\d\\mathbb{D}^{k}\\times 0$ and $0\\times \\d \\mathbb{D}^{4-k}$ are called the \\emph{attaching} and \\emph{belt} spheres of $h$, the image of $\\mathbb{D}^{k}\\times 0$ is called the \\emph{core} of $h$, see  \\cite[Fig.\\ 4.1]{GS_Kirby}. Smoothing corners gives a $4$-manifold $V\\cup_{\\phi}h$ which depends only on the attaching sphere of $h$ and, if $k=2$, on the integer called \\emph{framing} of $h$, which equals the linking number of its attaching circle and its push-off along a transverse vector field \\cite[4.1, 4.5]{GS_Kirby}. For example, blowing up is the same as attaching a $2$-handle with framing $-1$ \\cite[p.\\ 150]{GS_Kirby}.\n\nAny $4$-manifold $V$ can be obtained from $\\mathbb{D}^{4}$ by attaching handles of increasing index. There are two ways of modifying  such handle-decomposition without changing the diffeomorphism type of $V$. First, if the attaching sphere of a $k$-handle meets the belt sphere of a $(k-1)$-handle transversally in a single point, these two handles can be \\emph{canceled} \\cite[4.2.9]{GS_Kirby}. Second, a $2$-handle $h$ can be \\emph{slid over} a $2$-handle $h_{0}$ by pushing the attaching circle of $h$ through the belt circle of $h_0$ \\cite[4.2.10]{GS_Kirby}. We use the same letter to denote a handle before and after the slide.\n\nA handle-decomposition of $V$ is encoded by a \\emph{Kirby diagram}, which is a (decorated) link in $\\mathbb{S}^{3}$, i.e.\\ in the boundary of the initial $0$-handle. A $2$-handle is represented by its attaching circle with a framing coefficient. Attaching a $1$-handle to $V$ is the same as drilling a tubular neighborhood of a properly embedded $\\mathbb{D}^{2}\\subseteq V$ \\cite[p.\\ 168]{GS_Kirby}, so it is represented by a circle $\\d\\mathbb{D}^{2}\\subseteq \\d V$, with a dot to distinguish it from a $2$-handle. All dotted circles in a Kirby diagram form an unlink. Once $1$- and $2$-handles are attached, there is usually (e.g.\\ if $V$ is closed or $\\d V$ is connected and $V$ is simply connected, which is our case) a unique way to attach the remaining ones \\cite[p.\\ 148]{GS_Kirby}, so they are not drawn. We use the same letters for handles and corresponding knots in the diagram. \n\nFor example, a diagram consisting of one unknot with coefficient $e$ represents a $\\mathbb{D}^{2}$-bundle over $\\mathbb{S}^{2}$ of Euler number $e$ \\cite[Fig.\\ 4.20]{GS_Kirby}. The diagram in Figure \\ref{fig:handles} without the leftmost and rightmost circle represents $\\mathbb{T}^{2}\\times \\mathbb{D}^2$, see \\cite[Fig.\\ 4.36]{GS_Kirby}. Indeed, attaching the $1$-handles gives $(\\mathbb{T}^{2}\\setminus \\{\\mbox{disk}\\}) \\times \\mathbb{D}^{2}$, whose fundamental group is freely generated by loops based in the initial $0$-handle and going along the cores of the $1$-handles. The $2$-handle, attached along their commutator, caps off the puncture.\n\nHandle cancellation and sliding is represented on the Kirby diagram as follows. First, if a dotted circle meets only one non-dotted circle, once, then the corresponding $1$- and $2$-handle cancel \\cite[Fig.\\ 5.38]{GS_Kirby}. Second, sliding a $2$-handle $t$ over $t_0$ replaces $t$ by its band-sum with a parallel copy $\\tilde{t}_0$ of $t_0$ \\cite[Fig.\\ 5.4]{GS_Kirby}. More precisely, choose an arc in $t$ and $\\tilde{t}_0$ and connect their endpoints such that the obtained \\enquote{square} bounds a band which is disjoint from the rest of the diagram. Recall that $\\ell k(t_0,\\tilde{t}_0)$ is the framing of $t_0$. The new framing of $t$ is the sum of framings of $t$ and $t_0$ plus $2\\ell k(t,t_0)$ \\cite[p.\\ 142]{GS_Kirby}. It is important to remember that the orientations of $t$ and $\\tilde{t}_0$ should match. We denote by $-t$ the knot $t$ taken with opposite orientation. The components of a standard Hopf link drawn as in the left of Figure \\ref{fig:added_torus} are always meant to have counterclockwise orientation and linking number $+1$. \n\n\\@startsection{subsection}{3{Handle decompositions and proof of Theorem \\ref{thm:diff}}\\label{sec:diff_proof}\n\n\\begin{notation}\\label{not:d}\n\tLet $S=S_{p_{1},p_{2}}$ be as in Construction \\ref{constr}. For $j\\in \\{1,2\\}$ put $d_{j}=\\deg p_{j}+1$, $T_{j,d_j}=L_{j,0}$, and denote by $T_{j,i}$, $i\\in \\{1,\\dots, d_{j}-1\\}$ the $i$-th component of the twig $T_{j}$ meeting $A_{j}$, see Figure \\ref{fig:Skl}. Let $\\mathrm{Tub}(D)$ be the tubular neighborhood of $D$ in $X$ constructed in \\cite{Mumford-surface_singularities}. Put $M=\\d \\mathrm{Tub}(D)$ as in Section \\ref{sec:graph_manifolds} and $V=X\\setminus \\mathrm{int}\\, \\mathrm{Tub}(D)$. Clearly, $S\\simeq_{\\mathrm{diffeo}} \\mathrm{int}\\,  V$. We may and will  assume $d_1\\leq d_2$.\n\\end{notation}\n\n\n\\begin{prop}\\label{prop:handles}\n\tThe $4$-manifold $V$ is obtained from $\\mathbb{T}^{2} \\times \\mathbb{D}^2$ by attaching two $2$-handles with framings $-d_{1}$, $-d_{2}$ along the standard generators of $\\pi_{1}(\\mathbb{T}^{2})$. Its Kirby diagram is given by Figure \\ref{fig:handles}.\n\\end{prop}\n\\begin{proof} \n\tLet $\\phi$ be as in Construction \\ref{constr}. Let $\\mathbb{D}\\subseteq \\mathbb{C}^{2}\\subseteq \\P^1\\times \\P^1$ be a closed disk of radius $2$ centered at $(0,0)$, so that the base points $(0,1)$, $(1,0)$ of $\\phi^{-1}$ belong to $\\mathrm{int}\\, \\mathbb{D}$. Put $X_0:= \\phi^{-1}(\\mathbb{D})\\subseteq X$, $\\phi_{0}:= \\phi|_{X_0}\\colon X_0\\to \\mathbb{D}$ and $\\ll_{i,j}'=\\ll_{i,j}\\cap \\mathbb{D}$. Figure \\ref{fig:link} shows  $\\d \\phi_{0}^{-1}(\\ll_{1,0}' \\cup \\ll_{2,0}')$, where the attaching circles of the $2$-handles are drawn together with their framings.\n\t \t \\begin{figure}[ht]\n\t \t \t\\includegraphics[scale=0.35]{figures\/link}\n\t \t \t\\caption{Underlying disks of $L_{1,0}$, $L_{2,0}$ and handles attached to $\\mathbb{C}^{2}$ by $\\phi_{0}$.}\n\t \t \t\\label{fig:link}\n\t \t \\end{figure}\n\t \t \t \n\t To construct $V$, we need to remove from $X_0$ a tubular neighborhood of $\\phi_{0}^{-1}(\\ll_{1,0}' \\cup \\ll_{2,0}')\\setminus (A_{1}\\cup A_{2})$. To do this, we first remove from the $0$-handle all disks whose boundaries are drawn in Figure \\ref{fig:link} in bold, and then we remove the cores of the corresponding $2$-handles (everything suitably thickened).\n\t \n\t By \\cite[p. 214]{GS_Kirby}, the first step gives a Kirby diagram obtained from Figure \\ref{fig:link} as follows. Replace each circle corresponding to $T_{j,i}$ for $j\\in \\{1,2\\}$, $i\\in \\{1,\\dots, d_{j}\\}$ by a dotted circle $t_{j,i}'$, and each twist corresponding to $T_{j,i+1}\\cap T_{j,i}$ (resp.\\ $L_{0,1}\\cap L_{0,2}$) by a $0$-framed $2$-handle $h_{j,i}$ (resp.\\ $h$), linked with the dotted circles as in Figure \\ref{fig:added_torus}. We denote by $a_{j}$ the $2$-handle corresponding to $A_{j}$. \n\t \t\n\t \t\n\t \t \\begin{figure}[ht]\n\t \t \t\\includegraphics[scale=0.35]{figures\/added_torus}\n\t \t \t\\caption{Intersection of two removed disks, see \\cite[Fig.\\ 6.27]{GS_Kirby}.}\n\t \t \t\\label{fig:added_torus}\n\t \t \\end{figure}\n\tBy \\cite[p.\\ 224]{GS_Kirby}, the next step amounts to attaching a $2$-handle along a parallel copy of each removed circle, together with a certain $3$-handle. In our case, for each $T_{j,i}$, $j\\in \\{1,2\\}$, $i\\in \\{1,\\dots, d_{j}-1\\}$ we attach a $(-2)$-framed $2$-handle $t_{j,i}$ such that $\\ell k(t_{j,i},t_{j,i}')=-2$ and $\\ell k(t_{j,i},t_{j,i+1}')=\\ell k(t_{j,i},t_{j,i-1}')=1$, where we put $t_{j,0}'=a_{j}$. The resulting Kirby diagram is shown in  Figure \\ref{fig:V} (where we only draw the part corresponding to $L_{1,0}+L_{2,0}+T_{2}+A_{2}$ and skip the subscript \\enquote{$2,$}: the other part is analogous). \n\\begin{figure}[ht]\n\t\\includegraphics[scale=0.32]{figures\/V_half}\n\t\\caption{Right half of the Kirby diagram for $V=X\\setminus \\mathrm{Tub}(D)$.}\n\t\\label{fig:V}\n\\end{figure}\n\t\n\tWe will simplify this diagram by handle slides. First, we slide $t_{j,1}$ over $a_{j}$, see Figure \\ref{fig:first_slide}.   After this slide, $t_{j,1}$ unlinks from $a$, meets $t_{j,1}'$ once with $\\ell k(t_{j,1},t_{j,1}')=-1$ and changes framing to $-1$, while the rest of the diagram does not change. \n\t\\begin{figure}[htbp]\n\t\t\\begin{tabular}{ccccc}\n\t\t\t\\begin{subfigure}[t]{0.22\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.3]{figures\/ta_1}\n\t\t\t\\end{subfigure}\n\t\t\t&\n\t\t\t\\raisebox{1.2cm}{\\scalebox{1.5}{$\\rightsquigarrow$}}\n\t\t\t&\n\t\t\t\\begin{subfigure}[t]{0.22\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.3]{figures\/ta_2}\n\t\t\t\\end{subfigure}\n\t\t\t&\n\t\t\t\\raisebox{1.2cm}{\\scalebox{1.5}{$\\sim$}}\n\t\t\t&\n\t\t\t\\begin{subfigure}[t]{0.22\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.3]{figures\/ta_3}\n\t\t\t\\end{subfigure}\n\t\t\\end{tabular}\n\t\t\\caption{Sliding $t_{j,1}$ over $a_j$.}\n\t\t\\label{fig:first_slide}\n\t\\end{figure}\t\t\n\tApplying Lemma \\ref{lem:slides}\\subref{fig:ind_0} below inductively to pairs $(t_0,t)=(t_{j,i},t_{j,i+1})$ for $i=1,\\dots,d_j-1$, we obtain a diagram in Figure \\ref{fig:V_disjoint}. \n\t \t \\begin{figure}[ht]\n\t \t \t\\includegraphics[scale=0.32]{figures\/V_disjoint}\n\t \t \t\\caption{Right half of the Kirby diagram after sliding $t_{i}$ over $t_{i-1}$ for $i=1,2,\\dots, d_{2}-1$}\n\t \t \t\\label{fig:V_disjoint}\n\t \t \\end{figure}\t\n \t \n\tNow by Lemma \\ref{lem:slides}\\subref{fig:sliding_torus} each $h_{i,j}$ can be slid to a $0$-framed unknot disjoint from the rest of the diagram. Since $\\pi_{1}(V)=\\{1\\}$ by Proposition \\ref{prop:htp}, \\cite[p.\\ 148]{GS_Kirby} shows that all $h_{j,i}$'s cancel with all $3$-handles (recall that there are $d_1+d_2-2$ of them, attached while removing the cores of $T_{j,i}$'s). Applying Lemma \\ref{lem:slides}\\subref{fig:a_0} inductively  to pairs $(a,t)=(t_{j,i},t_{j,i+1})$, $i=1,\\dots,d_j-1$ gives Figure \\ref{fig:handles}.\n\\end{proof}\n\nIn the above proof, we used the following standard exercise in handle slides, cf.\\ \\cite[p.\\ 142]{GS_Kirby}.\n\n\n\\begin{lem}\\label{lem:slides}\n\tLet $\\ll_{1}$, $\\ll_{2}$ be links in the left and right of one of the Figures \\ref{fig:slides}\\subref{fig:ind_0}-\\subref{fig:a_0}. Define $\\ll\\subseteq \\ll_1$ as $t_0$ in \\subref{fig:ind_0}, $t+h$ in \\subref{fig:sliding_torus} and $t+c_0+a$ in \\subref{fig:a_0}. Let $\\mathscr{k}$ be a Kirby diagram containing $\\ll_{1}$ in such a way that $\\ll$ is disjoint from $\\mathscr{k}-\\ll_{1}$. Then $\\mathscr{k}$ is equivalent to a diagram obtained from $\\mathscr{k}$ by replacing $\\ll_{1}$ with $\\ll_{2}$.\n\\end{lem}\n\t \t \\begin{figure}[ht]\n\t \t \t\\begin{tabular}{cc}\n\t \t \t\t\\begin{subfigure}[t]{0.55\\textwidth}\n\t \t \t\t\t\\includegraphics[scale=0.17]{figures\/ind}\n\t \t \t\t\t\\caption{Sliding $t$ over $t_0$}\n\t \t \t\t\t\\label{fig:ind_0}\n\t \t \t\t\\end{subfigure}\t\n\t \t \t\t&\n\t \t \t\t\\setcounter{subfigure}{2}\n\t \t \t\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t \t \t\t\t\\includegraphics[scale=0.27]{figures\/a}\n\t \t \t\t\t\\caption{Sliding $t$ over $a$ and canceling $(a,c_0)$}\n\t \t \t\t\t\\label{fig:a_0}\n\t \t \t\t\\end{subfigure}\n\t \t \t\t\\\\\n\t \t \t\t\\multicolumn{2}{c}{\n\t \t \t\t\\setcounter{subfigure}{1}\n\t \t \t\t\t\\begin{subfigure}[t]{\\textwidth}\n\t \t \t\t\t\t\\centering\n\t \t \t\t\t\t\\includegraphics[scale=0.35]{figures\/sliding_torus}\n\t \t \t\t\t\t\\caption{After adding and subtracting $t$, a $2$-handle $h$ like in Figure \\ref{fig:added_torus} is ready to be canceled with a $3$-handle}\n\t \t \t\t\t\t\\label{fig:sliding_torus}\n\t \t \t\t\t\\end{subfigure}\n\t \t \t\t}\n\t \t \t\\end{tabular}\n\t \t \t\\caption{Lemma \\ref{lem:slides}: handle slides applied inductively in the proof of Proposition \\ref{prop:handles}}\n\t \t \t\\label{fig:slides}\n\t \t \\end{figure}\n\\begin{proof}\n\t\\subref{fig:ind_0}\n\tIn order to have a clear picture, we draw $t_0$ rotated around a horizontal line, so that both $t_0$ and $t$ go under the bottom part of $c_0$, see Figure \\ref{fig:ind}. We draw a parallel copy of $t_0$ in such a way that it twists over $t_0$ near that crossing, and cut this twist to make a band-sum with $t$. Now the part of $t$ on the right of the vertical diameter of $c$ can be isotoped to the latter, which proves the claim. \n\t\\\n\t\\begin{figure}[ht]\n\t\t\\begin{subfigure}{\\textwidth}\n\t\t\t\\begin{tabular}{ccccc}\n\t\t\t\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\t\t\t\\includegraphics[scale=0.19]{figures\/ind_1}\n\t\t\t\t\\end{subfigure}\n\t\t\t\t&\n\t\t\t\t\\raisebox{1.2cm}{\\scalebox{1.5}{$\\rightsquigarrow$}}\n\t\t\t\t&\n\t\t\t\t\\begin{subfigure}[t]{0.27\\textwidth}\n\t\t\t\t\t\\includegraphics[scale=0.19]{figures\/ind_2}\n\t\t\t\t\\end{subfigure}\n\t\t\t\t&\n\t\t\t\t\\raisebox{1.2cm}{\\scalebox{1.5}{$\\sim$}}\n\t\t\t\t&\n\t\t\t\t\\begin{subfigure}[t]{0.27\\textwidth}\n\t\t\t\t\t\\includegraphics[scale=0.19]{figures\/ind_3}\n\t\t\t\t\\end{subfigure}\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Sliding $t$ over $t_0$}\n\t\t\t\\label{fig:ind}\n\t\t\\end{subfigure}\n\t\\smallskip\n\n\t\\begin{subfigure}{\\textwidth}\n\t\t\\begin{tabular}{ccccc}\n\t\t\t\\begin{subfigure}[t]{0.27\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.35]{figures\/st_1}\n\t\t\t\\end{subfigure}\n\t\t\t&\n\t\t\t\\raisebox{1cm}{\\scalebox{1.5}{$\\rightsquigarrow$}}\n\t\t\t&\n\t\t\t\\begin{subfigure}[t]{0.27\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.35]{figures\/st_2}\n\t\t\t\\end{subfigure}\n\t\t\t&\n\t\t\t\\raisebox{1cm}{\\scalebox{1.5}{$\\sim$}}\n\t\t\t&\n\t\t\t\\raisebox{-.2cm}{\n\t\t\t\t\\begin{subfigure}[t]{0.27\\textwidth}\n\t\t\t\t\t\\includegraphics[scale=0.35]{figures\/st_3}\n\t\t\t\t\\end{subfigure}\n\t\t\t}\n\t\t\\end{tabular}\n\t\t\\caption{After sliding over $t$, $h$ becomes its parallel copy.}\n\t\t\\label{fig:st_proof}\n\t\\end{subfigure}\n\t\\smallskip\n\t\n\t\\begin{subfigure}{\\textwidth}\n\t\t\\begin{tabular}{ccccc}\n\t\t\t\\begin{subfigure}[t]{0.26\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.3]{figures\/a_1}\n\t\t\t\\end{subfigure}\n\t\t\t&\n\t\t\t\\raisebox{1.2cm}{\\scalebox{1.5}{$\\rightsquigarrow$}}\n\t\t\t&\n\t\t\t\\begin{subfigure}[t]{0.28\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.3]{figures\/a_2}\n\t\t\t\\end{subfigure}\n\t\t\t&\n\t\t\t\\raisebox{1.2cm}{\\scalebox{1.5}{$\\rightsquigarrow$}}\n\t\t\t&\n\t\t\t\\begin{subfigure}[t]{0.27\\textwidth}\n\t\t\t\t\\includegraphics[scale=0.3]{figures\/a_3}\n\t\t\t\\end{subfigure}\n\t\t\\end{tabular}\n\t\t\\caption{Canceling the rightmost $1\/2$-handle pair.}\n\t\t\\label{fig:sliding_a}\n\t\\end{subfigure}\n\t\t\\caption{Proof of Lemma \\ref{lem:slides}.}\n\t\t\\label{fig:sliding_lemma}\n\\end{figure}\n\n\t\n\t\\subref{fig:sliding_torus}\t As in \\subref{fig:ind_0}, we draw $t$ rotated so that both $h$ and $t$ go under the bottom part of $c_2$, and do the band-sum there, see Figure \\ref{fig:st_proof}. Then the bottom part of $h$ untangles from $c_1$, $c_2$, and becomes a parallel copy of $t$. Sliding it over $-t$ unlinks it from the rest of the  diagram.\n\t\n\t\\subref{fig:a_0} After sliding $t$ over $a$, it becomes disjoint from $c_0$ and meets $c$ as before, see Figure \\ref{fig:sliding_a}, where the box stands for $k$ twists. Moreover, $c_0$ now meets only $a$, once, so these handles cancel.\t\n\\end{proof}\t \t \n\t \t \n\n\\begin{proof}[Proof of Theorem \\ref{thm:diff}]\n\tRecall that by Proposition \\ref{prop:handles}, $S$ is diffeomorphic to an interior of a $4$-manifold with Kirby diagram as in Figure \\ref{fig:handles}, and by Corollary \\ref{cor:not_homeo} such $S$'s are not homeomorphic for different $\\{d_1,d_2\\}$. Therefore, it remains to prove that Kirby diagrams in Figures \\ref{fig:handles} and \\ref{fig:knot} are equivalent.\n\t\n\\begin{figure}[ht]\t\n\\begin{subfigure}{.8\\textwidth}\n\t\\begin{tabular}{ccc}\n\t\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\t\\includegraphics[scale=0.33]{figures\/h_1}\n\t\t\\end{subfigure}\n\t\t&\n\t\t\\raisebox{1.2cm}{\\scalebox{1.5}{$\\sim$}}\n\t\t&\n\t\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\t\\includegraphics[scale=0.33]{figures\/h_12}\n\t\t\\end{subfigure}\n\t\\end{tabular}\n\t\\caption{Adding $a_2-a_1$ to $h$.}\n\t\\label{fig:h}\n\\end{subfigure}\n\\begin{subfigure}{.8\\textwidth}\n\t\\begin{tabular}{ccc}\n\t\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\t\\includegraphics[scale=0.33]{figures\/h_2}\n\t\t\\end{subfigure}\n\t\t&\n\t\t\\raisebox{1.2cm}{\\scalebox{1.5}{$\\sim$}}\n\t\t&\n\t\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\t\\includegraphics[scale=0.33]{figures\/h_3}\n\t\t\\end{subfigure}\n\t\\end{tabular}\n\t\\caption{Subtracting $a_2-a_1$ from $h$ and canceling pairs $(c_j,a_j)$.}\n\t\\label{fig:hn}\n\t\\end{subfigure}\n\\caption{Proof of Theorem \\ref{thm:diff}\n\\label{fig:proof_diff}\n\\end{figure}\t\t\n\n\tLet $h$, $a_{j}$ be the $0$- and $(-d_{j})$-framed $2$-handles in Figure \\ref{fig:handles}, and let $c_{j}$ be the $1$-handle meeting $a_{j}$. We will add and subtract $a_2-a_1$ from $h$ so that the latter untangles from $c_1$, $c_2$. Then we cancel pairs $(c_j,a_j)$. This is done in Figure \\ref{fig:proof_diff}, which we now explain; see also \\cite[5.1.8(b)]{GS_Kirby} for the case $d_1=d_2=1$.\n\t\n\tWe start with Figure \\ref{fig:handles}, but draw it with $a_2$ rotated along a horizontal axis. Next, we draw parallel copies of $-a_{1}$ and $a_{2}$, where a box stands for $d_{j}$ twists. Then we band sum $h$ with those copies near the points where $h$ and $a_{j}$ go under $c_{j}$, see Figure \\ref{fig:h}. This move resolves nearby crossings of $h$ and $c_j$, leaving  $\\ell k(h,c_j)=-1$. Now we rotate $a_2$ again (this changes the sign in the box) and repeat the above move, but with reversed orientation of $a_j$, i.e.\\ we add $a_1$ and subtract $a_2$. As before, the crossings of $h$ with $c_j$ get resolved, so these $1$-handles cancel with $a_j$, see Figure \\ref{fig:hn}. After the first slide, $h$ twisted along $a_j$ in the boxes, so now it twists there over itself. A look at Figure \\ref{fig:hn} shows that we do obtain a knot as in Figure \\ref{fig:knot}. The framing of $h$ remains $0$ since we added and subtracted the same thing. \t\n\\end{proof}\n\n\\@startsection{subsection}{3{Computation of $\\pi_{1}^{\\infty}(S_{p_{1},p_{2}})$ and a relation with knot surgeries}\\label{sec:pi_1}\n\nProposition \\ref{prop:handles} allows us to compute the fundamental group of $V$ (which is trivial by Proposition \\ref{prop:htp}), and, more interestingly, that of $M$, which equals $\\pi_{1}^{\\infty}(S)$, see \\cite{Mumford-surface_singularities}.\n\n\\begin{prop}\\label{prop:pi_1}\nThe fundamental group at infinity of $S$ has a presentation\n\\begin{equation}\\label{eq:pi_1}\n\\pi_{1}^{\\infty}(S)=\n\\langle \\delta_{1},\\delta_{2},\\lambda\\ | \\ \\delta_{1}=[\\gamma_2,\\lambda^{-1}],\\ \\delta_{2}=[\\gamma_1,\\lambda],\\ [\\gamma_1,\\gamma_2]=1\\rangle, \\quad \\mbox{ where } \\gamma_{j}=\\delta_{j}^{d_j}.\n\\end{equation}\nMoreover, $\\gamma_j$ for $j\\in \\{1,2\\}$ is the attaching circle of the $(-d_j)$-framed $2$-handle in Figure \\ref{fig:handles}, and $\\lambda$ is the belt sphere of the $0$-framed one.\n\\end{prop}\n\\begin{proof}\n\tLet $\\gamma_{1},\\gamma_{2},\\lambda\\in \\pi_{1}(\\d (\\mathbb{T}^{2}\\times \\mathbb{D}^{2}))=\\pi_{1}(\\mathbb{T}^{3})$ be the standard generators. Proposition \\ref{prop:handles} shows that $M$ is obtained from $\\mathbb{T}^{3}$ by a surgery along $\\gamma_{1},\\gamma_{2}$ with coefficients $d_{1}$, $d_{2}$. The meridians of deleted tori are $\\mu_{1}:=[\\gamma_{2},\\lambda^{-1}], \\mu_{2}:= [\\gamma_{1},\\lambda]$. Now by van Kampen theorem, each surgery replaces the relation $\\mu_j=1$ in $\\pi_{1}(\\mathbb{T}^{3})$ by $\\mu_j=\\delta_j$, where $\\delta_j\\in \\pi_{1}(M)$ is the corresponding attaching circle, satisfying $\\delta_{j}^{d_j}=\\gamma_{j}$.\n\\end{proof}\n\n\\begin{rem}\\label{rem:pi_1-Mumford}\n\tThe presentation \\eqref{eq:pi_1} can also be obtained from the plumbing construction in \\cite{Mumford-surface_singularities}, see Section \\ref{sec:graph_manifolds}. If $D$ is a tree, \\cite{Mumford-surface_singularities} shows how to calculate $\\pi_{1}(M)$ in terms of fibers of the plumbed $\\mathbb{S}^{1}$-bundles, using van Kampen theorem. For our $D$, one needs to use the van Kampen theorem for groupoids \\cite{Brown_van-Kampen}, say, to glue $\\d \\mathrm{Tub}(L_{1,\\infty}+L_{2,\\infty})$ to the rest  of $\\d \\mathrm{Tub}(D)$ along two disjoint plumbing tori, each containing a base point. This gives the additional generator $\\lambda$, which is the sum of paths joining the two base points in each of the glued pieces. In this interpretation, $\\gamma_{j}$ is the fiber over $L_{j,0}$. Indeed, in the proof of Proposition \\ref{prop:handles} it was drawn parallel to the boundary of a disk corresponding to $T_{j,d_j-1}$, which goes once around $L_{j,0}$. In turn, $\\delta_{j}$ is a fiber over $T_{j,1}$, that is, a singular fiber of a lens space $L(1,d_{j})$ attached to $\\d \\mathrm{Tub}(L_{j,0})$.\n\\end{rem}\n\nIn the proof of Corollary \\ref{cor:not_homeo}, we have essentially seen the JSJ decomposition of $M$. We will now show how to see it from the above description of $M$, as a $0$-surgery on $K_{[2d_1,2d_2]}$. In order to settle the notation, we recall the basic definitions following \\cite{Hatcher_3M}.\n\nLet $N$ be a $3$-manifold. A surface $\\Sigma$ in $N$ is \\emph{incompressible} if every loop in $\\Sigma$ bounding a disk in $N$ bounds a disk in $\\Sigma$, too. We write $N|\\Sigma$ for a $3$-manifold (with boundary) obtained from $N$ by removing $\\Sigma$ together with some tubular neighborhood. \nAssume that $N$ is irreducible (see Section \\ref{sec:graph_manifolds}). Then $N$ has a collection $\\Sigma$ of incompressible tori such that each component of $N|\\Sigma$ is either a Seifert fiber space or has no incompressible tori. A minimal  such $\\Sigma$  (in the sense of inclusion) is unique up to an isotopy \\cite[Theorem 1.9]{Hatcher_3M}, and $N|\\Sigma$ is called a \\emph{JSJ-decomposition} of $N$. \n\nWe now recall the notation for Seifert fibered spaces following \\cite[\\S 2.1]{Hatcher_3M}. Let $B$ be a compact surface of genus $g$ with $r$ boundary components, and let $M_0\\to B_0$ be an $\\mathbb{S}^{1}$-bundle, where $B_0$ is $B$ with $k$ disks $D_{1},\\dots, D_{k}$ removed. The preimage $T_{i}\\subseteq M_{0}$ of $\\d D_{i}$ is a torus: we attach to it a solid torus $\\mathbb{D}^{2}\\times \\mathbb{S}^{1}$ in such a way that a meridian $0\\times \\mathbb{S}^{1}$ is attached to a loop in $T_{i}$ whose preimage in its universal cover is a line of slope $\\tfrac{p_{i}}{q_{i}}$. The resulting manifold is a \\emph{Seifert fibered space} $M(g,r;\\tfrac{p_{1}}{q_{1}},\\dots, \\tfrac{p_{k}}{q_k})$. It is a graph manifold. Its graph has one vertex corresponding to $M_0$ and $k$ maximal twigs, whose type is a continued fraction expansion of $\\tfrac{p_i}{q_i}$. This graph is normal whenever a Seifert fibration is unique, see \\cite[5.7]{Neumann-plumbing_graphs}. Few important exceptions, including lens spaces $\\mathbb{S}^3\/\\mathbb{Z}_{p}$, are listed in \\cite[Theorem 2.3]{Hatcher_3M}. Their normal graphs are described in \\cite[6.1]{Neumann-plumbing_graphs}.\n\nLet again $M=\\d \\mathrm{Tub}(D)$ be as in Notation \\ref{not:d}. The following result can be inferred from the computation of $\\Gamma(M)$ in Figure \\ref{fig:graphs} using the algorithm in \\cite[\\S 5]{Neumann-plumbing_graphs}. Nonetheless, we sketch a direct argument using the geometric description of incompressible tori in $M$ deduced in \\cite[Lemma 2.2]{BW_surgery} from \\cite{HT_inc}.\n\n\n\n\\begin{prop}[JSJ decomposition of $M$]\\label{prop:JSJ}\n\tLet $\\Sigma_{1},\\Sigma_{2}\\subseteq M$ be the plumbing tori at $L_{1,0}\\cap L_{2,0}$ and $L_{1,0}\\cap L_{2,\\infty}$, see Section \\ref{sec:graph_manifolds}. Then the JSJ decomposition of $M$ is\n\t\\begin{enumerate}\n\t\t\\item\\label{item:d_1>1} $M|(\\Sigma_{1}\\sqcup \\Sigma_2)=M(2,0;\\tfrac{1}{d_1})\\sqcup M(2,0;\\tfrac{1}{d_2})$ if $d_{1},d_{2}\\neq 1$,\n\t\t\\item\\label{item:d_1=1} $M|\\Sigma_1=M(2,0;\\tfrac{1}{d_2})$ if $d_{1}=1$ and $d_2>1$,\n\t\t\\item\\label{item:Moser} $M=M(0,0;\\tfrac{1}{2},\\tfrac{1}{3},\\tfrac{1}{6})$ if $d_1=d_2=1$.\n\t\\end{enumerate}\n\tLet $\\gamma_{j},\\delta_{j}$ be as in Proposition \\ref{prop:pi_1}. Then $\\delta_{j}$ is the singular fiber of $M(2,0;\\tfrac{1}{d_j})$ and $\\gamma_{j}$ is a general one.\n\\end{prop}\n\\begin{proof}\n\tBy Theorem \\ref{thm:diff}, $M$ is a $0$-surgery on $\\mathbb{S}^{3}$ along $K=K_{[2d_1,2d_2]}$. Because $K_{[2,2]}$ is just a trefoil, \\ref{item:Moser} follows from \\cite[Proposition 3.1]{Moser_knots} which completely describes surgeries on torus knots. Assume $d_2>1$. If $d_1>1$, too, then cutting along $\\Sigma_1\\sqcup \\Sigma_2$ in case \\ref{item:d_1>1} corresponds to removing the edges of the circular subgraph in Figure \\ref{fig:g_2}, and decorating the leftmost vertices with $r=2$, i.e.\\ puncturing the base spheres twice. If $d_1=1$ then cutting along $\\Sigma_1$ corresponds to removing the loop in Figure \\ref{fig:g_3} and adding boundary as before. This shows the equalities in \\ref{item:d_1>1}--\\ref{item:d_1=1}, and proves that $M$ and, if $d_1=1$, $M|\\Sigma_i$ for $i\\in \\{1,2\\}$ are not Seifert fibered: indeed, $\\Gamma(M|\\Sigma_i)$ is not one of the graphs in \\cite[5.7, 6.1]{Neumann-plumbing_graphs}. It remains to show that $\\Sigma_{1},\\Sigma_{2}\\subseteq M$ are incompressible.\n\t\n\tThe $3$-manifold $M$ is irreducible by \\cite[Theorem 2]{HT_inc} (it also follows from Lemma \\ref{lem:graphs}\\ref{item:irr}). Put $M_0=\\mathbb{S}^{3}\\setminus \\mathrm{Tub}(K)\\subseteq M$. Put $M_0=\\mathbb{S}^{3}\\setminus \\mathrm{Tub}(K)\\subseteq M$. By \\cite[Proposition 1(1) and the proof of Proposition 2]{HT_inc} there are, up to isotopy, at most two  incompressible surfaces $\\Sigma_0\\subseteq M_0$ such that the surfaces $\\Sigma\\subseteq M$ obtained by capping $\\Sigma_0$ are tori (see also proof of \\cite[Lemma 2.2]{BW_surgery}). Such $\\Sigma_0$ are denoted by $S_{1}(0)$ and $S_{1}(1)$ in \\cite{HT_inc}, and they have exactly one boundary component each. By \\cite[Proposition 1.5(a)]{Przytycki_inc} they are unknotted (see Definition 1.3 loc. cit), so by Theorem 1.4 loc.\\ cit, $\\Sigma$ remains incompressible. Moreover, by \\cite[Theorem 1(e)]{HT_inc} $S_1(0)$ is isotopic to $S_1(1)$ if and only if $d_1=1$. Thus our claim will follow once we show that $\\Sigma=\\Sigma_1,\\Sigma_2$ for $\\Sigma_0=S_1(0),S_1(1)$. \n\t\n\tTo this end, we recall the description of $S_{1}(0)$, $S_{0}(1)$ from \\cite[p.\\ 227, Fig.\\ 1]{HT_inc}. Consider Figure \\ref{fig:knot} as a knot in $\\mathbb{R}^{3}$, and let $\\mathscr{p}$ be a vertical plane between the \\enquote{twists}. It meets the knot in four points. Connect them to get a quadrilateral  $\\mathscr{q}\\subseteq \\mathscr{p}$: it is a $0$-handle for $S_{1}(0)$. Now attach to it two $1$-handles along the bands with $d_{1}$ and $d_{2}$ twists, respectively: this gives a punctured torus $S_{1}(0)$. The torus $S_{1}(1)$ is constructed the same way, with $\\mathscr{q}$ replaced by its complement in the compactification $\\bar{\\mathscr{p}}=\\mathbb{S}^{2}\\times \\{\\mathrm{pt}\\}\\subseteq (\\mathbb{S}^{2}\\times \\mathbb{R})\\sqcup \\{\\pm \\infty\\}=\\mathbb{S}^{3}$.\n\t\n\tThe loops based at $\\mathscr{q}$ and going around the $1$-handles of $S_1(0)$ are isotopic to the non-dotted circles in Figure \\ref{fig:handles}, which by Remark \\ref{rem:pi_1-Mumford} are fibers $\\gamma_j$ over $L_{j,0}$. Hence $S_{1}(0)$ caps to $\\Sigma_1$. To see the $1$-handles of $S_{1}(1)$, we need to change the base point from $\\mathscr{q}$ to its complement. The gluing described in Remark \\ref{rem:pi_1-Mumford} shows that this amounts to changing, say, $L_{2,0}$ to $L_{2,\\infty}$. This way we get $\\Sigma_2$, as claimed.\n\\end{proof}\n\n\\begin{rem}[A direct argument for minimality of $\\Sigma$]In the above proof, we have used the classification of graph manifolds to show that the collections $\\Sigma_1\\sqcup\\Sigma_2$ in \\ref{item:d_1>1} and $\\Sigma_1$  in \\ref{item:d_1=1} of incompressible tori decomposing $M$ to Seifert fibered spaces is minimal. However, this can be also seen directly, as follows.\n\t\n\nFirst, we note that since $\\Gamma(M|\\Sigma_{i})$ is a tree, one can use van Kampen theorem as in \\cite{Mumford-surface_singularities} to compute\n\\begin{equation}\\label{eq:pi_1-Sigma}\n\\pi_{1}(M|\\Sigma_{1})=\\langle \\delta_{1},\\delta_{2}|[\\gamma_1,\\gamma_2]=1 \\rangle, \\mbox{ where } \\delta_j\\mbox{ and } \\gamma_j=\\delta_{j}^{d_{j}}\\mbox{ are as in Remark \\ref{rem:pi_1-Mumford}.}\n\\end{equation}\n\n\nSuppose that $d_1,d_2>1$, but $M|\\Sigma_i$ is Seifert fibered for some $i\\in \\{1,2\\}$. Then by \\cite[Proposition 1.11]{Hatcher_3M}, the incompressible torus $\\Sigma_{3-i}\\subseteq M|\\Sigma_{i}$ is either horizontal or vertical. In the first case, it covers the base orbifold, which has two boundary components, so comparing Euler characteristics as done in \\cite[p.\\ 22]{Hatcher_3M} shows that $M|\\Sigma_i$ has no singular fibers, so $M|\\Sigma_{i}\\cong (\\mathbb{D}^{1}\\times \\mathbb{S}^{1})\\times \\mathbb{S}^{1}$, a contradiction with \\eqref{eq:pi_1-Sigma}. In the second case, cutting $M|\\Sigma_{i}$ further along $\\Sigma_{3-i}$  preserves the Seifert fibration: therefore by the equality in \\ref{item:d_1>1}, it restricts to the unique one on $M(2,0;\\tfrac{1}{d_{j}})$. Hence after gluing $\\Sigma_{3-i}$ back, the fiber of one fibration becomes a loop around the degenerate fiber of the other. This is a contradiction, because the former is central in $\\pi_{1}$, and the latter is not.\n\nSuppose now that $d_{2}>1$, but $M$ is Seifert fibered. Again, the incompressible torus $\\Sigma_{1}$ is either horizontal or vertical. In the first case, since $M|\\Sigma_{1}$ is connected, \\cite[p.\\ 22]{Hatcher_3M} implies that $M|\\Sigma_1\\cong \\Sigma_1\\times \\mathbb{D}^{1}$, a contradiction with \\eqref{eq:pi_1-Sigma}. In the second case, we obtain that $M|\\Sigma_{1}$ is a Seifert fiber space. We have shown that this is possible only if $d_{1}=1$, in which case the general fiber is represented by $\\gamma_{1}\\in \\pi_{1}(M)$. In particular, $\\langle \\gamma_{1}\\rangle \\subseteq \\pi_{1}(M)$ is a (nontrivial)  central subgroup. Hence by \\eqref{eq:pi_1} $\\delta_{2}=[\\gamma_{1},\\lambda]=1$ and $\\delta_{1}=[\\delta_{2}^{d_{2}},\\lambda^{-1}]=1$, so $\\gamma_{1}=1$; a contradiction.\n\\end{rem}\n\n\n\n\\section{$\\mathbb{C}^{*}$-fibrations and complete algebraic vector fields on $S_{p_1,p_{2}}$}\\label{sec:AAut}\n\nIn this section we prove Theorem \\ref{thm:AAut}\\ref{item:AAut_thm}. It follows from \\cite[Theorem 1.6]{KKL_avf} that a flow of any complete algebraic vector field on $S_{p_{1},p_{2}}$ preserves some $\\mathbb{C}^{*}$-fibration of $S_{p_{1},p_{2}}$ (see Proposition \\ref{prop:fibrations}\\ref{item:vf} for details). Therefore, to understand $\\mathrm{AAut}_{\\mathrm{hol}}(S_{p_{1},p_{2}})$ we need to classify all $\\mathbb{C}^{*}$-fibrations of $S_{p_{1},p_{2}}$. \n\nTheir degenerate fibers are affine curves with Euler characteristic at least $e_{\\mathrm{top}}(\\mathbb{C}^{*})$, see \\cite[III.1.8]{Miyan-OpenSurf}, i.e.\\ unions of $\\mathbb{C}^{*}$ and $\\mathbb{C}^{1}$. Therefore, we begin by classifying all curves in $S_{p_{1},p_{2}}$ which are isomorphic to $\\mathbb{C}^{1}$. This is done in Lemma \\ref{lem:C1}. In Lemma \\ref{lem:C*_lines} we show that any $\\mathbb{C}^{*}$-fibration of $S_{p_{1},p_{2}}$ comes from a $\\mathbb{C}^{*}$-fibration of a complement of suitably chosen pair of disjoint $\\mathbb{C}^{1}$'s; which by Lemma \\ref{lem:tori} turns out to be a torus $\\mathbb{C}^{*}\\times \\mathbb{C}^{*}$. The union of such tori is exactly the complement of $Z$ from Theorem \\ref{thm:AAut}\\ref{item:AAut_thm}.\n\n\\begin{notation}\\label{not:S}\n\tAs in Notation \\ref{not:d}, we put $d_j=\\deg p_{j}+1$ and write $S:= S_{p_{1},p_{2}}$. We use notation introduced in Construction \\ref{constr} and denote by $T_{j}$ the maximal $(-2)$-twig of $D$ meeting $A_{j}$. Additionally, we denote by $\\mathscr{c}=\\{x_1+x_2=1\\}\\subseteq \\P^{1}\\times \\P^{1}$ the curve of type $(1,1)$ passing through $(\\infty,\\infty)$ and both base points $(1,0)$, $(0,1)$ of $\\phi^{-1}$, and by $C:=\\phi^{-1}_{*}\\mathscr{c}$ its proper transform on $X$. \n\t\n\tIt would be sometimes convenient to work with explicit equations of $S$. To write them, denote by $\\hat{p}_{j}=t^{d_{j}-1}p_{j}(t^{-1})$, $j\\in \\{1,2\\}$ the polynomial obtained from $p_{j}$ by writing its coefficients in the opposite order. Note that $\\hat{p}_{j}(0)=1$ because $p_{j}$ is monic. Now $S\\subseteq \\mathbb{C}^{4}=\\operatorname{Spec}\\mathbb{C}[x_1,x_2,y_1,y_2]$ is given by\n\t\\begin{equation}\\label{eq:S}\n\ty_1x_1^{d_1}=x_2-\\hat{p}_1(x_1),\\quad y_2x_2^{d_2}=x_1-\\hat{p}_2(x_2).\n\t\\end{equation}\n\tNote that $S$ is in fact a surface in $\\mathbb{C}^{3}$ because, say, $x_1$ can be computed from the second equation.\n\\end{notation}\n\n\n\n\n\\begin{lem}[{$\\mathbb{C}^{1}$'s in $S$, cf.\\ \\cite[Lemma 8.1]{Kal-Ku1_AL-theory}}]\\label{lem:C1}\n\tA curve $\\Gamma\\subseteq X$ satisfies $\\Gamma\\cap S\\cong \\mathbb{C}^{1}$ if and only if one of the following holds:\n\t\\begin{enumerate}\n\t\t\\item\\label{item:A} $\\Gamma=A_{j}$ for some $j\\in \\{1,2\\}$. Then $\\Gamma^{2}=-1$ and $\\Gamma\\cdot D=\\Gamma\\cdot T_j=1$.\n\t\t\\item\\label{item:L} $\\Gamma=L_{j,1}$ and $p_{j}=t^{d_{j}-1}$ for some $j\\in \\{1,2\\}$. Then $\\Gamma^{2}=-d_{j}$, $\\Gamma\\cdot D=\\Gamma\\cdot L_{3-j,\\infty}=1$ and $\\Gamma\\cdot A_{3-j}=1$. \n\t\t\\item\\label{item:C} $\\Gamma=C$ and $p_{j}=1$ or $t^{d_{j}-2}(t-1)$ for both $j\\in \\{1,2\\}$. Then $\\Gamma^{2}=2-d_{1}-d_{2}$, $\\Gamma\\cdot D=2$, $\\Gamma$ meets $D$ at $L_{1,\\infty}\\cap L_{2,\\infty}$ and $\\Gamma\\cdot A_{j}=1$ for $j\\in \\{1,2\\}$.\n\t\\end{enumerate}\n\tMoreover, $A_{1}\\cdot A_{2}=0$, $L_{1,1}\\cdot L_{2,1}=1$ and $A_{j}\\cdot L_{j,1}=C\\cdot L_{j,1}=0$ for $j\\in \\{1,2\\}$.\n\\end{lem}\n\\begin{proof}\n\tClearly, $A_{j}\\cap S\\cong \\mathbb{C}^{1}$ and $A_{j}^{2}=-1$ as in \\ref{item:A}. We will now check when $L_{j,1}\\cap S \\cong \\mathbb{C}^{1}$. Because $\\ll_{j,1}$ meets $\\phi_{*}D$ transversally on $\\ll_{3-j,0}$ and $\\ll_{3-j,\\infty}$, we have  $L_{j,1}\\cdot(D+A_{3-j})=2$ and $L_{j,1}$ meets $L_{3-j,\\infty}$, so $L_{j,1}\\cap S\\cong \\mathbb{C}^{1}$ if and only if $L_{j,1}$ meets $A_{3-j}$, that is, all $d_{j}$ blowups over $\\ll_{j,1}\\cap \\ll_{3-j,0}$ touch the image of $L_{j,1}$ (hence $L_{j,1}^{2}=\\ll_{j,1}^{2}-d_{j}=-d_{j}$). The proper transform of $\\ll_{j,1}$ after the first blowup meets the exceptional curve at the point with coordinate $(x_{3-j}-1)\/x_{j}=0$ (see Construction \\ref{constr}), so the next blowup, if occurs, touches it if and only if the coefficient near $t^{d_{j}-2}$ in $p_{j}$ is zero. Because these two curves meet transversally, the infinitely near points on the next proper transforms will have coordinate zero, too. Following Construction \\ref{constr} we conclude that $L_{j,1}\\cap S\\cong \\mathbb{C}^{1}$ if and only if $p_{j}=t^{d_{j}-1}$, as in \\ref{item:L}.\n\n\tTo check when $C\\cap S\\cong \\mathbb{C}^{1}$, we argue in a similar way. We have $C\\cdot(D+A_{1}+A_{2})=4$ and $C$ meets $D$ at $L_{1,\\infty}\\cap L_{2,\\infty}$, so $C\\cap S\\cong \\mathbb{C}^{1}$ if and only if all the blowups in the decomposition of $\\phi$ touch the image of $C$. After the first blowup over $\\ll_{j,1}\\cap \\ll_{3-j,0}$ the proper transform of $\\mathscr{c}$ meets the exceptional curve at a point of coordinate $(x_{3-j}-1)\/x_{j}=1$, so all the remaining blowups touch it if and only if $p_{j}=1$ (i.e.\\ there are no more blowups) or $p_{j}=t^{d_{j}-1}-t^{d_{j}-2}$, since as before the next infinitely near points will have coordinate zero on the exceptional curves. In this case, $C^{2}=\\mathscr{c}^{2}-(d_{1}+d_{2})=2-d_{1}-d_{2}$, as in \\ref{item:C}.\n\t\\smallskip\n\t\n\tIt remains to show that if $\\Gamma\\cap S\\cong \\mathbb{C}^{1}$ for some curve $\\Gamma\\subseteq X$ then $\\Gamma$ equals $A_{j}$, $L_{1,j}$ or $C$. Consider first the case when $\\Gamma$ is vertical for the $\\P^{1}$-fibration induced by $|L_{j,\\infty}|$, so $\\Gamma\\subseteq \\phi^{*}\\ll_{j,t}$ for some $t\\in \\mathbb{C}^{1}$ because $\\Gamma\\not\\subseteq D$. If $t\\not\\in\\{0,1\\}$ then $\\Gamma\\cap S\\cong \\mathbb{C}^{*}$, which is false. If $t=0$ then $\\Gamma=A_{j}$ because $\\phi^{*}\\ll_{j,0}=L_{j,0}+T_{j}+A_{j}$. If $t=1$ then $\\Gamma=L_{1,j}$ or $A_{3-j}$ because $(\\phi^{*}\\ll_{j,1})_{\\mathrm{red}}=L_{j,1}+T_{3-j}+A_{3-j}$.\n\t\n\tConsider now the case when $\\Gamma$ is horizontal for both $\\P^{1}$-fibrations induced by $|L_{j,\\infty}|$, $j\\in \\{1,2\\}$. Then $\\Gamma$ meets both $L_{j,\\infty}$, so the point $\\Gamma\\cap D$ equals $L_{1,\\infty}\\cap L_{2,\\infty}$. Since $\\Gamma$ is disjoint from $L_{1,0}+L_{2,0}=[1,1]$, the linear system of the latter induces a $\\P^{1}$-fibration for which $\\Gamma$ is vertical. The curves $L_{j,\\infty}$ are $1$-sections for this $\\P^{1}$-fibration, so $\\Gamma\\cdot L_{j,\\infty}=1$. It follows that $\\phi(\\Gamma)\\subseteq \\P^{1}\\times \\P^{1}$ is of type $(1,1)$, passes through $(\\infty,\\infty)$, and because $\\Gamma\\cdot L_{j,0}=0$, $\\phi(\\Gamma)$ passes through the base point of $\\phi^{-1}$ on $\\ll_{j,0}$, that is, through $(0,1)$ for $j=1$ and $(1,0)$ for $j=2$. Thus $\\phi(\\Gamma)=\\mathscr{c}$, as claimed.\n\\end{proof}\n\n\\begin{lem} \\label{lem:C*_lines}\n\tFor any $\\mathbb{C}^{*}$-fibration of $S$ one can find on $S$ two disjoint, vertical curves isomorphic to $\\mathbb{C}^{1}$, whose closures $\\Gamma_{1},\\Gamma_{2}\\subseteq X$ satisfy $\\Gamma_{i}^{2}\\geq -1$, with strict inequality if $\\Gamma_{i}\\cdot D\\geq 2$.\n\\end{lem}\n\\begin{proof}\n\tFix a $\\mathbb{C}^{*}$-fibration $f$ of $S$. Let $\\alpha_{1}\\colon X''\\to X$ be the minimal resolution of the base points of $f$ on $X$, and let $\\alpha_{2}\\colon X''\\to X'$ be the contraction of all vertical $(-1)$-curves which are superfluous in the subsequent images of $(\\alpha_{1}^{*}D)_{\\mathrm{red}}$. Put $\\alpha=\\alpha_{2}\\circ\\alpha_{1}^{-1}$ and $D'=((\\alpha^{-1})^{*}D)_{\\mathrm{red}}$.\n\t\n\tDenote the circular subdivisor of $D$ by $R$. Then $\\alpha_{1}^{*}R$ contains a unique circular subdivisor $R''$. Because all fibers are trees, $R''$ contains a horizontal component, say $H$. Suppose $R''-H$ is vertical. Then it is connected and meets $H$ twice, so $H$ is a $2$-section, in fact $H=D''_{\\mathrm{hor}}$; a contradiction with Lemma \\ref{lem:untwisted}. Therefore, $R''$ contains two $1$-sections. In particular, $R''_{\\mathrm{hor}}=D''_{\\mathrm{hor}}$, so $\\alpha_{1}$ is a composition of inner blowups with respect to the images of $R''$, hence it does not touch $\\alpha_{1}^{*}(D-R)$. Because the latter contains no $(-1)$-curves and meets $R''$ in branching components of $D''$, $\\alpha_{2}$ does not touch it, either. In other words, $\\alpha$ is a composition of inner blowups on the images of $R''$ and does not touch the twigs of $D''$.\n\t\n\tIn particular, $D'_{\\mathrm{hor}}$ consists of two $1$-sections, both contained in $R'=(\\alpha_{2})_{*}R''$. \n\t\n\tConsider the case when $D'\\vert$ has a $(-1)$-curve, say $\\Gamma$, and let $F$ be the fiber containing it. Because $D'$ has no superfluous vertical $(-1)$-curves, $\\beta_{D'}(\\Gamma)\\geq 3$, so $\\Gamma\\subseteq R'$ meets a maximal twig $T$ of $D'$. Because $\\alpha^{-1}$ does not touch $T$, we get $\\alpha^{-1}_{*}(T+\\Gamma)=T_{j}+L_{j,0}=[(2)_{d_{j}},1]$ for some $j\\in \\{1,2\\}$. Since  $\\beta_{F_{\\mathrm{red}}}(\\Gamma)\\leq 2$, $\\Gamma$ meets $R'_{\\mathrm{hor}}$. The latter consists of $1$-sections, so $\\Gamma$ has multiplicity $1$ in $F$, and thus $\\Gamma$ is a tip of $F$. Because $F$ contracts to a $0$-curve, it follows that $F=[1,(2)_{d_{j}},1]$. The $(-1)$-curve $\\Gamma'=F-T-\\Gamma$ satisfies $\\Gamma'\\cdot D'=1+(F-T-\\Gamma)\\cdot D'_{\\mathrm{hor}}=1$, so $A=\\alpha^{-1}_{*}\\Gamma'$ satisfies $A\\cdot D=A\\cdot T_{j}=1$, thus $A=A_{3-j}$ by Lemma \\ref{lem:C1}. It follows that $F=\\alpha^{*}(A_{j}+T_{j}+L_{3-j,0})$, so our $\\P^{1}$-fibration is in fact induced by a pullback of $|L_{3-j,\\infty}|$. Therefore, $\\alpha=\\mathrm{id}$ and the curves $A_{j}\\cap S$ and $A_{3-j}\\cap S$ are as required.\n\t\n\tConsider the case when  $D'\\vert$ contains no $(-1)$-curves. Because $R'$ is circular, $R'_{\\mathrm{hor}}$ has $2-b_{0}(R'\\vert)\\in \\{0,1,2\\}$ nodes. Let $F_r$ be a fiber passing through a node $r\\in R'_{\\mathrm{hor}}$. Then $(F_{r}\\cdot D'_{\\mathrm{hor}})_{r}=2$, so because each connected component of $D'\\vert$ meets $D'_{\\mathrm{hor}}$, $F_{r}\\cap D'=\\{r\\}$. Because $X'\\setminus D'\\cong S$ is affine, $F_{r}\\cap (X'\\setminus D')\\cong \\mathbb{C}^{1}$. The curve $F_{r}$ meets each $1$-section in $D'_{\\mathrm{hor}}$ once, so the above description of $\\alpha$ shows that $\\alpha_{2}$ touches $(\\alpha_{2}^{-1})_{*}F_{r}$ at most once. Therefore,  $(\\alpha^{-1}_{*}F_{r})^{2}\\geq F_{r}^{2}-1=-1$, and the equality holds if and only if the proper transform of $F_{r}$ is touched by $\\alpha_{2}$ and not by $\\alpha_1$, in which case $\\alpha^{-1}_{*}F_{r}\\cdot D=1$. Hence $F_{r}\\cap S$ is as in the statement of the lemma. \n\t\n\tSimilarly, if $A\\subseteq X'$ is a vertical $(-1)$-curve such that $A\\cdot D'=1$, then $A\\not\\subseteq D'$ by assumption, and  $\\alpha_{2}$ does not touch $(\\alpha_{2}^{-1})_{*}A$, hence $A\\cap S$ is as in the statement, and clearly $A$ is disjoint from any $F_{r}$. Therefore, it remains to find $b_{0}(R'\\vert)$ disjoint vertical $(-1)$-curves $A\\subseteq X'$ such that $A\\cdot D'=1$.\n\t\n\tAssume that every connected component $V$ of $R'\\vert$ is contained in a fiber $F_{V}\\neq V$. Let $A_V\\subseteq F_V$ be a $(-1)$-curve. Then $A_V\\not\\subseteq D'$, $A_V\\cdot D'_{\\mathrm{hor}} \\leq (F_V-V)\\cdot D'_{\\mathrm{hor}}=0$ and $A_V\\cdot D'\\vert=1$ because $F_V$ is a tree, so $A_V\\cdot D'=1$. This ends the proof because  $A_V\\cap A_{V'}=\\emptyset$ for $V\\neq V'$.\n\t\n\tThus we can assume that $R'\\vert$ contains a fiber. We have $3\\leq \\#R'\\leq \\#D'=\\rho(X')$ by Lemma \\ref{lem:S0}. Hence $X'$ contains a degenerate fiber $F$. Let $A$ be a $(-1)$-curve in $F$, so $A\\not\\subseteq D'$. \n\t\n\tSuppose that $F\\subseteq D'\\vert+A$. Then $A$ is a unique $(-1)$-curve in $F$, so it has multiplicity $\\mu\\geq 2$ in $F$. Denote our $\\mathbb{C}^{*}$-fibration of $S$ by $f$. Because $D'$ contains a fiber, $f(S)\\subseteq\\mathbb{C}^{1}$, and we can assume that $f(A\\cap S)=\\{0\\}$. Because $\\mathbb{C}[S]$ is a UFD, $A|_{S}$ is a divisor of a regular function, say $g$. Then $f\/g^{\\mu}\\in \\mathbb{C}[S]^{*}=\\mathbb{C}^{*}$, so $f=\\lambda g^{\\mu}$ for some $\\lambda\\in \\mathbb{C}^{*}$. A general fiber of such $f$ is reducible, a contradiction.\n\t\n\tSuppose $L\\cdot D'\\geq 2$ for every $(-1)$-curve $L\\subseteq F$. Then $2\\leq L\\cdot D'\\leq b_0(F\\cap D')$ because $F$ is a tree, and $b_0(F\\cap D')\\leq F\\cdot D'_{\\mathrm{hor}}=2$ because $D'$ is connected. It follows that $A$ is a unique $(-1)$-curve in $F$, and meets two connected components of $F\\cap D'$.\u00a0Because $F$ meets $D'_{\\mathrm{hor}}$ in components of multiplicity $1$, it follows that $F$ is a chain meeting $D'$ in tips contained in $D'\\vert$. Hence $F\\subseteq D'\\vert+A$; a contradiction.\n\t\n\tThus we can assume $A\\cdot D'=1$. If $b_0(R'\\vert)<2$ then the lemma follows, so we can assume $b_0(R'\\vert)=2$. Lemma \\ref{lem:S0}\\ref{item:units} implies that $D'$ contains exactly one fiber, so some connected component $V$ of $R'\\vert$ is contained in a degenerate fiber $F$ as above. Recall that $A\\subseteq F$ is a $(-1)$-curve such that $A\\not\\subseteq D'$,  $A\\cdot D'=1$ and $F\\not\\subseteq D'\\vert+A$.\n\t\n\tFor every component $L$ of $F-D'\\vert$ we have $L\\cdot D'_{\\mathrm{hor}}\\leq (F'-V)\\cdot D'_{\\mathrm{hor}}=0$, so since $F$ is a tree, $L\\cdot D'=1$ and all such $L$'s are disjoint. If two of them are $(-1)$-curves then the lemma follows. Hence we can assume that $A$ is a unique $(-1)$-curve in $F$. Put $T=0$ if $A$ meets $V$, otherwise denote by $T$ the maximal twig of $D'$ meeting $A$. If $T\\neq 0$ then $\\alpha^{-1}$ does not touch $A+T$, hence by Lemma \\ref{lem:C1}, $A+T=\\alpha_{*}(A_{j}+T_{j})=[1,(2)_{d_{j}}]$ for some $j\\in \\{1,2\\}$. Moreover, if $T$ meets a curve $L\\not\\subseteq D'$ with $L\\cdot D'=1$ then Lemma \\ref{lem:C1} gives $L=A$. Hence $T\\cdot(F_{\\mathrm{red}}-A-V)=0$. In any case, let $\\tau$ be the contraction of $A+T$. Then $\\tau$ does not touch $F_{\\mathrm{red}}-V$, and the fiber $\\tau_{*}F$ has a unique $(-1)$-curve, contained in $\\tau_{*}V$. Perform further  inner contractions with respect to the images of the chain $\\tau_{*}V$ until one of the components of the image of $F$ meeting a $1$-section in the image of $D'_{\\mathrm{hor}}$ becomes a $(-1)$-curve. Then the latter has multiplicity $1$ in the image of $F$, so  it is not its unique $(-1)$-curve. It follows that the image of $F$ is of type $[1,1]$ and both its components are in the image of $V$. Hence $F\\subseteq D'\\vert+A$; a contradiction. \t\n\\end{proof}\n\nIn the following, we fix coordinates on $S=S_{p_{1},p_{2}}$, treating it as a closed subset of $\\mathbb{C}^{4}$ given by \\eqref{eq:S}.\n\n\\begin{lem}[Open tori in $S$]\\label{lem:tori}\n\tFix an open subset $U\\subseteq S$ and a ring homomorphism  $\\sigma\\colon \\mathbb{C}[x_1,x_2,y_1,y_2]\\to \\mathbb{C}[v_{1}^{\\pm 1},v_{2}^{\\pm 1}]$ as in \\ref{item:AA}--\\ref{item:LC} below. Then $\\sigma$ induces an isomorphism $U\\cong \\mathbb{C}^{*}\\times \\mathbb{C}^{*}$.\n\t\\begin{enumerate}\n\t\t\\item\\label{item:AA} $U=S\\setminus (A_1+A_2)$ and \n\t\t$\\sigma(x_{j})=v_{j}$, $\\sigma(y_{j})= (v_{3-j}-\\hat{p}_{j}(v_{j}))v_j^{-d_{j}}$ for $j\\in \\{1,2\\}$;\n\t\t\\item\\label{item:AL} In case $p_{3-j}=1$ for some $j\\in \\{1,2\\}$:\n\t\t $U=S\\setminus (A_{j}+L_{j,1})$ and\n\t\t \\begin{equation*}\n\t\t \\sigma(x_{j})=v_{j},\\quad\n\t\t \\sigma(x_{3-j})=(v_{j}-1)v_{3-j}^{-1},\\quad\n\t\t \\sigma(y_{j})=(v_{j}-1-\\hat{p}_{j}(v_{j})v_{3-j})\\cdot (v_{j}^{d_j}v_{3-j})^{-1},\\quad\n\t\t \\sigma(y_{3-j})=v_{3-j};\n\t\t \\end{equation*}\n\t\t\\item\\label{item:LC} In case $p_{1}=p_{2}=1$:\n\t\t $U=S\\setminus (C+L_{j,1})$ for some $j\\in \\{1,2\\}$ and\n\t\t \\begin{equation*}\n\t\t \\sigma(x_{j})=-(v_{2}+1)v_{1}^{-1},\\quad\n\t\t \\sigma(x_{3-j})=-(v_{1}+v_{2}+1)\\cdot (v_{1}v_{2})^{-1},\\quad\n\t\t \\sigma(y_{j})=(v_{1}+1)v_{2}^{-1},\\quad\n\t\t \\sigma(y_{3-j})=v_{2}.\n\t\t \\end{equation*}\n\t\\end{enumerate}\nThe volume form $\\tfrac{dv_{1}}{v_{1}}\\wedge \\tfrac{dv_{2}}{v_{2}}$ on $U$ extends, up to a sign, to a volume form $\\tfrac{dx_{1}}{x_{1}}\\wedge \\tfrac{dx_{2}}{x_{2}}$.\n\\end{lem}\n\\begin{proof}\n\tWe check that by a direct computation that $\\sigma$ is zero on the ideal of $S$, so it gives a morphism $\\mathbb{C}^{*}\\times \\mathbb{C}^{*}\\to S$. It has a rational inverse, given by\n\t\\begin{equation*}\n\t\\mbox{\\ref{item:AA} }v_{j}\\mapsto x_{j},\\ j\\in \\{1,2\\};\\quad \n\t\\mbox{\\ref{item:AL} }v_{j}\\mapsto x_{j}, v_{3-j}\\mapsto y_{3-j};\\quad\n\t\\mbox{\\ref{item:LC} }v_{1}\\mapsto y_{1}y_{2}-1, v_{2}\\mapsto y_{3-j}.\n\t\\end{equation*}\n\tThis inverse is regular on the complement of:\n\t\\begin{equation*}\n\t\\mbox{\\ref{item:AA} }\\{x_{1}x_{2}=0\\}=(A_{1}+A_{2})|_{S};\\quad \n\t\\mbox{\\ref{item:AL} }\\{x_{j}y_{3-j}=0\\}=(A_{j}+L_{j,1})|_{S};\\quad\n\t\\mbox{\\ref{item:LC} }\\{y_{3-j}(1-y_{1}y_{2})=0\\}=(L_{j,1}+C)|_{S};\n\t\\end{equation*}\t\n\tand sends $\\tfrac{dv_{1}}{v_{1}}\\wedge \\tfrac{dv_{2}}{v_{2}}$ to $\\pm \\tfrac{dx_{1}}{x_{1}}\\wedge \\tfrac{dx_{2}}{x_{2}}$. This proves the lemma. \n\\end{proof}\n\\begin{rem*}\n\tThe formulas $\\{y_{3-j}=0\\}=L_{j,1}|_{S}$, $\\{y_{1}y_{2}=1\\}=C|_{S}$ used in the above proof do \\emph{not} hold for general $p_{1},p_{2}$, but they do hold under the assumptions of \\ref{item:AL} and \\ref{item:LC}, respectively.\n\\end{rem*}\t\n\n\n\\begin{prop}[{Complete algebraic vector fields on $S_{p_{1},p_{2}}$, cf.\\ \\cite[Theorem 19]{Kal-Ku1_AL-theory}}]\\label{prop:fibrations}\\ \n\t\\begin{enumerate}\n\t\t\\item\\label{item:fibr} Every $\\mathbb{C}^{*}$-fibration of $S$ restricts to a $\\mathbb{C}^{*}$-fibration of one of the open tori $U\\subseteq S$ from Lemma \\ref{lem:tori}\n\t\t\\item\\label{item:extend} For any $U\\subseteq S$ from Lemma \\ref{lem:tori} and any $a_{j}\\geq d_{j}$, $j\\in \\{1,2\\}$, the complete vector field $\\nu_{a_{1},a_{2}}=v_{1}^{a_{1}}v_{2}^{a_{2}}(a_{2}v_{1}\\tfrac{\\d}{\\d v_{1}}-a_{1}v_{2}\\tfrac{\\d}{\\d v_{2}})$ on $U$ extends to a  complete vector field on $S$.\n\t\t\\item\\label{item:vf} Every complete algebraic vector field on $S$ restricts to a complete vector field on some torus $U\\subseteq S$ from Lemma \\ref{lem:tori}. In particular, it has divergence zero with respect to the volume form $\\tfrac{dx_1}{x_1}\\wedge \\tfrac{dx_2}{x_2}$.\n\t\t\\item\\label{item:AAut} For each open torus $U\\subseteq S$ from Lemma \\ref{lem:tori}, the group $\\mathrm{AAut}_{\\mathrm{hol}}(S)\\cap \\mathrm{AAut}_{\\mathrm{hol}}(U)$ acts on $U$ $m$-transitively for every $m$.\n\t\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n\t\\ref{item:fibr} Fix a $\\mathbb{C}^{*}$-fibration $p$ of $S$ and let $\\Gamma_{1},\\Gamma_{2}\\subseteq X$ be as in Lemma \\ref{lem:C*_lines}. Then $p$ restricts to a $\\mathbb{C}^{*}$-fibration of $U=S\\setminus (\\Gamma_{1}+\\Gamma_{2})$. We claim that $U$ is as in Lemma \\ref{lem:tori}. Because for $i\\in \\{1,2\\}$ $\\Gamma_{i}\\cap S\\cong \\mathbb{C}^{1}$, $\\Gamma_{i}$ is one of the curves from Lemma \\ref{lem:C1}. If $\\{\\Gamma_{1},\\Gamma_{2}\\}=\\{A_{1},A_{2}\\}$ then $U$ is as in Lemma \\ref{lem:tori}\\ref{item:AA}. Assume that $\\Gamma_{1}=C$. Because $C\\cdot D=2$, by Lemma \\ref{lem:C*_lines} $0\\leq \\Gamma_{1}^{2}=2-d_1-d_2$, so $d_1=d_2=1$, i.e.\\ $p_1=p_2=1$. In this case, $C\\cap S$ meets $A_j \\cap S$ for both $j\\in \\{1,2\\}$, so $\\Gamma_{2}=L_{1,j}$ for some $j\\in \\{1,2\\}$ and therefore $U$ is as in Lemma \\ref{lem:tori}\\ref{item:LC}. We are left with the case when $\\Gamma_{1}=L_{1,j}$ for some $j\\in \\{1,2\\}$ and $\\Gamma_{2}\\neq C$. As before, the condition $\\Gamma_{1}^{2}\\geq -1$ from Lemma \\ref{lem:C*_lines} implies by Lemma \\ref{lem:C1}\\ref{item:L} that $d_{j}=1$, i.\\ e.\\ $p_{j}=1$.  In this case, $L_{1,j}\\cap S$ meets $A_{3-j}\\cap S$ and $L_{3-j,1}\\cap S$, so $\\Gamma_{2}=A_{j}$ or $C$. The second case is excluded by assumption, so the first case holds and $U$ is as in Lemma \\ref{lem:tori}\\ref{item:AL}.\n\t\n\t\\ref{item:extend} follows from a direct computation, see \\cite[Corollary 2]{Andersen-tori}.\n\t\n\t\\ref{item:vf} Part \\ref{item:extend} implies that complete algebraic vector fields on $S$ do not share a rational first integral. Hence by \\cite[Theorem 1.6]{KKL_avf}, for any complete algebraic vector field $\\xi$ on $S$ there is a $\\mathbb{C}^{1}$- or $\\mathbb{C}^{*}$-fibration of $S$ preserved by the flow of $\\xi$. Because $\\kappa(S)=0$, $S$ admits no $\\mathbb{C}^{1}$-fibrations, hence $\\xi$ preserves one of the $\\mathbb{C}^{*}$-fibrations from \\ref{item:fibr}. In each case, the complement of $U$ is a union of vertical curves isomorphic to $\\mathbb{C}^{1}$, so the flow of $\\xi$ cannot send them to non-degenerate fibers in $U$. It follows that $\\xi$ restricts to a complete vector field on $U$. In particular, by \\cite[Corollary 1]{Andersen-tori} it preserves the volume form $\\tfrac{dv_{1}}{v_{1}}\\wedge \\tfrac{dv_{2}}{v_{2}}$, where $v_{1},v_{2}$ are coordinates on $U\\cong \\mathbb{C}^{*}\\times \\mathbb{C}^{*}$. Lemma \\ref{lem:tori} implies that $\\xi$ preserves $\\tfrac{dx_{1}}{x_{1}}\\wedge \\tfrac{dx_{2}}{x_{2}}$, too.\n\t\n\t\\ref{item:AAut} Choose integers $a_{ij}$, $i,j\\in \\{1,2\\}$ such that $a_{ij}\\geq d_{j}$ and $\\det [a_{ij}]=1$, and put $\\sigma_{i}=\\nu_{a_{i1},a_{i2}}$, $f_{i}=v_{1}^{a_{i1}}v_{2}^{a_{i2}}\\in\\ker \\sigma_{i}\\setminus \\ker \\sigma_{3-i}$. Now the result follows from  \\cite[Propostion 8.9]{Kal-Ku1_AL-theory}. Indeed, although that result is formulated in loc.\\ cit.\\ for $G=\\mathrm{AAut}_{\\mathrm{hol}}(U)$, the proof actually shows $m$-transitivity for the group generated by the elements of the flows of $q(f_{i})\\cdot \\sigma_{i}$ for $q\\in \\mathbb{C}[t]$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:AAut}]Part \\ref{item:Aut_thm} was shown in Corollary \\ref{cor:st}\\ref{item:Aut}. For \\ref{item:AAut_thm}, note that the open subset $S\\setminus Z$ of $S$ is precisely the union of all $U\\subseteq S$ from Lemma \\ref{lem:tori}. By Proposition \\ref{prop:fibrations}\\ref{item:vf}, $S\\setminus Z$ is fixed by $\\mathrm{AAut}_{\\mathrm{hol}}(S)$. On the other hand, by Proposition \\ref{prop:fibrations}\\ref{item:AAut} each $U\\subseteq S$ from Lemma \\ref{lem:tori} is contained in a single orbit of $\\mathrm{AAut}_{\\mathrm{hol}}(S)$. Because every two such $U$, being open and dense in $S$, have nonempty intersection, they are all contained in the same orbit of $\\mathrm{AAut}_{\\mathrm{hol}}(S)$, which therefore equals $S\\setminus Z$. Proposition \\ref{prop:fibrations}\\ref{item:AAut} implies that $\\mathrm{AAut}_{\\mathrm{hol}}(S)$ acts on this orbit $m$-transitively for every $m$, as claimed.\n\\end{proof}\n\n\\begin{rem}[Density property for $S_{p_{1},p_{2}}$]\n\tIn view of Theorem \\ref{thm:AAut}\\ref{item:AAut_thm}, it is natural to ask for an even stronger property than $m$-transitivity of $\\mathrm{AAut}_{\\mathrm{hol}}(S)$, namely for the \\emph{algebraic density property} (ADP), see \\cite{Kal-Ku1_AL-theory}. Recall that a smooth affine variety $X$ has ADP if the Lie algebra $\\operatorname{VF_{alg}}(X)$ of algebraic vector fields on $X$ coincides with the Lie algebra $\\operatorname{Lie_{alg}}(X)$ generated by complete ones. Since in our case all vector fields preserve $Z$, one should rather ask if the \\emph{relative} ADP holds, see \\cite{KLM_relative-density}, namely, if there exists $l\\geq 0$ such that $I^{l}\\operatorname{VF_{alg}}(S)\\subseteq \\operatorname{Lie_{alg}}(S)$, where $I$ is the ideal of $Z$. This, however, is not true since by Lemma \\ref{lem:tori}\\ref{item:vf} all elements of $\\operatorname{Lie_{alg}}(S)$ preserve the volume form $\\omega=\\tfrac{dx_1}{x_1}\\wedge \\tfrac{dx_2}{x_2}$. Nonetheless, one could ask for the relative \\emph{volume} density property, defined as follows (see \\cite{KK_avdp}). Let $X$ be a normal variety with a volume form $\\omega$, and let $Y\\subseteq X$ be a closed subset containing $\\operatorname{Sing} X$. Denote by $\\operatorname{VF_{alg}^{\\omega}}(X,Y)$ the Lie algebra of those algebraic vector fields on $X$ which vanish on $Y$ and preserve the volume form $\\omega$; and by  $\\operatorname{Lie_{alg}^{\\omega}}(X,Y)$ the Lie algebra generated by complete vector fields in $\\operatorname{VF_{alg}^{\\omega}}(X,Y)$. We say that $X$ has an algebraic volume density property (AVDP) relative to $Y$ if there exists $l\\geq 0$ such that $I^{l}\\operatorname{VF_{alg}^{\\omega}}(X,Y)\\subseteq \\operatorname{Lie_{alg}^{\\omega}}(X,Y)$, where $I$ is the ideal of $Y$. In this setting, one could ask the following:\n\\begin{question}\n\tLet $Z\\subseteq S_{p_{1},p_{2}}$ be as in Theorem \\ref{thm:AAut}\\ref{item:AAut_thm}. Does $S_{p_{1},p_{2}}$ have AVDP relative to $Z$?\n\\end{question}\nFor $p_{1}=p_{2}=1$ (so $Z=\\emptyset$), the answer to this question is positive \\cite[Theorem 6]{KK_avdp}\n\\end{rem}\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn $1882$, M. Noether claimed the following statement which was later proven by Lefschetz:\nFor $d \\ge 4$, a very general smooth degree $d$ surface $X$ in $\\mathbb{P}^3$ has Picard number $\\rho(X)=1$.\nThis motivates the definition of the \\emph{Noether-Lefschetz locus}, denoted by $\\NL_d$ parametrizing the \nspace of smooth degree $d$ surfaces $X$ in $\\mathbb{P}^3$ with $\\rho(X)>1$.\nOne of the interesting problems is to understand the geometry of the Noether-Lefschetz locus. By the Lefschetz $(1,1)$-theorem, we can look at an irreducible\ncomponent of the Noether-Lefschetz locus locally as a Hodge locus (see \\cite[\\S $5$]{v5} for more details).\nIn particular, denote by $U_d \\subseteq \\mathbb{P}(H^0(\\mathbb{P}^3, \\op))$ \nthe open subscheme parametrizing smooth projective hypersurfaces in $\\mathbb{P}^3$ of degree $d$.\nLet $\\mathcal{X} \\xrightarrow{\\pi} U_d$ be the corresponding universal family. For a given $F \\in U_d$, denote by $X_F$ the surface $X_F:=\\pi^{-1}(F)$. \nLet $X \\in U_d$ and $U \\subseteq U_d$ be a simply connected neighbourhood of $X$ in $U_d$ (under the analytic topology).\nThen, $\\pi|_{\\pi^{-1}(U)}$ induces a variation of Hodge structure $(\\mathcal{H}, \\nabla)$ on $U$, where $\\mathcal{H}:=R^2\\pi_*\\mathbb{Z} \\otimes \n\\mathcal{O}_U$ and $\\nabla$ is the Gauss-Manin connection. Note that $\\mathcal{H}$ defines a local system on $U$ whose fiber over \na point $F \\in U$ is $H^2(X_F,\\mathbb{Z})$. Consider a non-zero element $\\gamma_0 \\in H^2(X_F,\\mathbb{Z}) \\bigcap H^{1,1}(X_F,\\mathbb{C})$\nsuch that $\\gamma_0  \\not= c_1(\\mathcal{O}_{X_F}(k))$ for $k \\in \\mathbb{Z}_{>0}$. \nThis defines a section $\\gamma \\in (\\mathcal{H} \\otimes \\mathbb{C})(U)$. Let $\\overline{\\gamma}$ be the image\nof $\\gamma$ in $\\mathcal{H}\/F^2(\\mathcal{H} \\otimes \\mathbb{C})$. The Hodge loci corresponding to $\\gamma$, denoted $\\NL(\\gamma)$ is then defined as\n\\[ \\NL(\\gamma):=\\{G \\in U | \\overline{\\gamma}_G=0\\},\\]\nwhere $\\overline{\\gamma}_G$ denotes the value at $G$ of the section $\\overline{\\gamma}$. For an irreducible component $L \\subset \\NL_d$ and $X \\in L$, general, we can find $\\gamma \\in H^{1,1}(X,\\mb{Z}):= \nH^2(X,\\mb{Z}) \\bigcap H^{1,1}(X,\\mb{C})$ such that $\\overline{\\NL(\\gamma)}=L$ (the closure taken in $U_d$ under Zariski topology).\n\nOne of the first results in this direction is due to Green, Griffiths, Voisin and others (\\cite{M3, GH, v2}) which states that \nfor an irreducible component $L$ of the Noether-Lefschetz locus, that for $d \\ge 4$,\n\\[d-3 \\le \\codim (L,U_d) \\le \\binom{d-1}{3}.\\] \nThe upper bound follows easily from the fact that $\\dim H^{2,0}(X)=\\binom{d-1}{3}$ for any $X \\in U_d$ (see \\cite[\\S $6$]{v5}).\nWe say that $L$ is a \\emph{general} component if $\\codim L=\\binom{d-1}{3}$ and \\emph{special} otherwise. \nIt was proven by Ciliberto, Harris and Miranda \\cite{ca1} that for $d \\ge 4$, \nthe Noether-Lefschetz locus has infinitely many general components and the union of these components is Zariski dense in $U_d$. \nThe guiding principle of much work in the area has been the expectation that special components should be due to the presence of low degree curves.\nVoisin \\cite{v3} and Green \\cite{M3} independently \nproved that for $d \\ge 5$, $\\codim L=d-3$ if and only if $L$ parametrizes surfaces of degree $d$ containing a line. \nIf $d-3<\\codim L \\le 2d-7$ then $\\codim L=2d-7$ and $L$ parametrizes the surfaces containing a conic.\nOtwinowska \\cite{A1} proved that for an integer $b>0$ and $d \\gg b$ if $\\codim L \\le bd$ then $L$ parametrizes surfaces containing a curve of \ndegree at most $b$.\n\n\n\nFor $r \\ge 3$, we define the level $r$-\\emph{Noether-Lefschetz locus}, denoted $\\NL_{r,d}$ to be the space parametrizing surfaces with \nPicard number greater than or equal to $r$. It has been conjectured by Griffiths and Harris in \\cite{GH} that for $r < d$, an irreducible component of $\\NL_{r,d}$ is \nof codimension greater than or equal to $(r-1)(d-3)-\\binom{r-3}{2}$. Furthermore, the component of $\\NL_{r,d}$ parametrizing\nsurfaces containing $r-1$ coplanar lines is of this codimension. In this article we prove (in Theorem \\ref{gh6}) that:\n\\begin{thm}\\label{ite1}\n Let $r \\ge 3$ and $d \\gg  r$. Let $L$ be an irreducible component of $\\NL_{r,d}$. Then, $\\codim L \\ge (r-1)(d-3)-\\binom{r-3}{2}$.\nFurthermore, there exists a component $L$ of $\\NL_{r,d}$ of this codimension parametrizing surfaces containing $r-1$ coplanar lines.\n\\end{thm}\n\nThe techniques used to prove this result is a combination of deformation theory and Hodge theory. Instead of looking at the Hodge locus corresponding to a Hodge class, we study the Hodge locus corresponding to\na $\\mb{Z}$-module of Hodge classes. We then use a result due to Otwinowska, \\cite[Theorem $1$]{ot}, to show that if the codimension of an irreducible component $L$ of $\\NL_{r,d}$\nis less than or equal to $(r-1)(d-3)-\\binom{r-3}{2}$, then for a general $X \\in L$ there exists a lattice $\\Lambda \\subset H^{1,1}(X,\\mb{Z})$ generated by classes of curves of degree less than or\nequal to $r-1$ such that $L$ is locally of the form $\\NL(\\Lambda)$ (see Proposition \\ref{gh14}), where $\\NL(\\Lambda)$ is the intersection of $\\NL(\\gamma)$ for all $\\gamma \\in \\Lambda$,\n$\\bigcap_{\\gamma \\in \\Lambda} \\NL(\\gamma)$.\n\n\n\nWe now use the theory of semi-regularity as introduced in \\cite{b1} to reduce the problem to a question in flag Hilbert schemes. \nFirst to fix some notations, for a Hilbert polynomial $P$ for some curve $C$ in $\\p3$, we denote by $H_P$ the corresponding Hilbert scheme, parametrizing curves (schemes with pure dimension $1$) with \nHilbert polynomial $P$.\nThroughout this article we denote by $Q_d$ the Hilbert polynomial of a degree $d$\nsurface in $\\p3$. We denote by $H_{P,Q_d}$ the corresponding flag Hilbert scheme parametrizing pairs $(C,X)$ such that $C \\in H_P, X \\in H_{Q_d}$ and $C \\subset X$.\nA curve $C$ on a smooth surface in $\\p3$ is said to be \\emph{semi-regular} if $H^1(\\mo_X(C))=0$. \nWe prove that\n\\begin{thm}\n Let $X$ be a smooth surface in $\\p3$ of degree $d$ and $C \\subset X$, a semi-regular curve with Hilbert polynomial, say $P$. For any irreducible component, $L'$ of $\\ov{\\NL([C])}$\n (the closure is taken in $U_d$ under Zariski topology)\n there exists an irreducible component $H'$ of $H_{P,Q_d}$ containing the pair $(C,X)$ such that $\\pr_2(H')_{\\red}$ \n coincides with $L'_{\\red}$, where $\\pr_2$ is \n the second projection map from $H_{P,Q_d}$ to $H_{Q_d}$. In particular, if $C$ is reduced, connected and $d \\ge \\deg(C)+4$ then this holds true and the irreducible component $H'$ is uniquely determined by $L'$.\n\\end{thm}\nSee Theorem \\ref{dim} and Lemma \\ref{hf11} for the precise statements and its proof. Using this proof we further show that under the above \nbound on the codimension of $L$, the lattice will infact be generated by classes of lines (see Lemma \\ref{gh4}). \nFinally, we do a computation in Proposition \\ref{gh13}, to determine the ``arrangement'' of these lines which would \nhelp us determine the component with the correct codimension.\n\n\n\n\n\n{\\bf{Acknowledgement:}} I would like to thank Prof. R. Kloosterman for reading the preliminary version of this article and several helpful discussions. \n\n\\section{Introduction to Noether-Lefschetz locus}\n\n\\begin{para}\n In this section we recall the basic definitions of Noether-Lefschetz locus. See \\cite[\\S $9, 10$]{v4} and \\cite[\\S $5, 6$]{v5} for a detailed presentation of the subject.\n\\end{para}\n\n\n\\begin{note}\\label{n1}\nBy a \\emph{component} of $\\NL_d$, we mean an irreducible component. By a \\emph{surface} we always mean a smooth surface in $\\mathbb{P}^3$.\nDenote by $Q_d$ the Hilbert polynomial of degree $d$ surfaces in $\\p3$. Given, a Hilbert polynomial $P$, denote by $H_P$ the corresponding Hilbert scheme and by $H_{P,Q_d}$ the corresponding flag Hilbert scheme.\nAlso, for a point $u \\in U_d$, denote by $X_u$ the fiber $\\pi^{-1}(u)$.\n\\end{note}\n\n\\begin{note}\\label{nl1}\n Let $X \\in U_d$ and $\\mo_X(1)$, the very ample line bundle on $X$ determined by the closed immersion $X \\hookrightarrow \\p3$ arising (as in \\cite[II.Ex.$2.14$(b)]{R1}) from the graded homomorphism\nof graded rings $S \\to S\/(F_X)$, where $S=\\Gamma_*(\\mo_{\\p3})$ and $F_X$ is the defining equations of $X$.\nDenote by $H_X$ the very ample line bundle $\\mo_X(1)$. \nNote that a very ample line bundle on $X_u$ for any $u \\in U$ remains very ample in the family $\\mc{X}$, hence the corresponding cohomology class remains of type $(1,1)$ \nin $\\mc{X}$.\n\\end{note}\n\n     \n     \\begin{para}\n     Let $X$ be a surface.\n      The Lefschetz hyperplane section theorem implies, \\[H^{2}(X,\\mb{C})\\cong H^{2}(X,\\mb{C})_{\\prim} \\oplus \\mb{C}H_X,\\]\n      where $H_X$ is the very ample line bundle on $X$ and $H^2(X,\\mb{C})_{\\prim}$ is the primitive cohomology. \n      This gives us a natural projection map from $H^{2}(X,\\mb{C})$ to $H^{2}(X,\\mb{C})_{\\prim}$. For $\\gamma \\in H^{2}(X,\\mb{C})$, denote by $\\gamma_{\\prim}$ the image of $\\gamma$ under this morphism.\n      Since the very ample line bundle $H_X$ remains of type $(1,1)$ in the family $\\mc{X}$, we can therefore conclude that\n        $\\gamma \\in H^{1,1}(X)$ remains of type $(1,1)$ if and only if $\\gamma_{\\prim}$ remains of type $(1,1)$. In particular,\n      $\\NL(\\gamma)=\\NL(\\gamma_{\\prim})$.\n     \\end{para}\n     \n\\begin{defi} \nWe now discuss the tangent space to the Hodge locus, $\\NL(\\gamma)$.\nWe know that the tangent space to $U$ at $X$, $T_XU$ is isomorphic to $H^0(\\N_{X|\\p3})$.\nThis is because $U$ is an open subscheme of the Hilbert scheme $H_{Q_d}$, the tangent space of which at the point $X$ is simply\n$H^0(\\N_{X|\\p3})$. \nGiven the variation of Hodge structure above, we have (by Griffith's transversality) the differential map:\n\\[\\overline{\\nabla}:H^{1,1}(X) \\to \\mathrm{Hom}(T_XU,H^2(X,\\mathcal{O}_X))\\]induced by the Gauss-Manin connection.\nGiven $\\gamma \\in H^{1,1}(X)$ this induces a morphism, denoted $\\overline{\\nabla}(\\gamma)$ from $T_XU$ to $H^2(\\mo_X)$.\nThe tangent space at $X$ to $\\NL(\\gamma)$ is then defined to be $\\ker(\\overline{\\nabla}(\\gamma))$.\n\\end{defi}\n\n\\begin{para}\\label{gr0}\n The boundary map \\[\\rho:H^0(\\N_{X|\\p3}) \\to H^1(\\T_X)\\] arising from the long exact sequence associated to the \n short exact sequence:\n \\[0 \\to \\T_X \\to \\T_{\\p3}|_X \\to \\N_{X|\\p3} \\to 0\\]\n is called the \\emph{Kodaira-Spencer} map. The morphism $\\overline{\\nabla}(\\gamma)$ is related to the Kodaira-Spencer map as we will see below.\n\\end{para}\n\n\\begin{para}\\label{c9}\n Note that there exists a natural cup product morphism, \n \\[H^1(X,\\T_X) \\otimes H^{1}(X,\\Omega^1_X) \\xrightarrow{\\bigcup} H^2(X,\\mathcal{O}_X).\\]\n For $\\gamma \\in H^1(\\Omega_X^1)$ this induces a morphism, denoted $\\bigcup \\gamma$, from $H^1(\\T_X)$ to $H^2(\\mo_X)$.\n We then have the following result in Hodge theory (see \\cite[Theorem $10.21$]{v4}):\n\\end{para}\n\n\\begin{lem}\\label{a3}\nThe differential map $\\overline{\\nabla}(\\gamma)$ conincides with the following:\n\\[T_XU\\cong H^0(\\N_{X|\\p3}) \\xrightarrow{\\rho} H^1(\\T_X) \\xrightarrow{\\bigcup \\gamma} H^2(\\mo_X).\\]\n\\end{lem}\n\n\\section{Hodge locus and Hilbert flag schemes}\n\n\\begin{para}\n In this section we define what is a semi-regular map. We then briefly study Hodge locus for a family of smooth projective\n surfaces in $\\p3$ and show how it is related to certain Hilbert flag schemes. More specifically, we shall study \n the Hodge locus corresponding to certain effective algebraic cycles which will be semi-regular. For such classes we will see that\n the Hodge locus ``coincides'' with a component of a flag Hilbert scheme. We elaborate on the details in this section.\n\\end{para}\n\n\n\\subsection{Semi-regularity map and tangent space to Hodge locus}\n\n\\begin{para}\nWe start with the definition of a semi-regular curve.\n Let $X$ be a surface and $C \\subset X$, a curve in $X$. Since $X$ is smooth, $C$ is local complete intersection in $X$.\n This gives rise to the short exact sequence:\\[0 \\to \\mo_X(-C) \\to \\mo_X \\to i_*\\mo_C \\to 0,\\]where $i$ is the natural inclusion\n morphism from $C$ into $X$. Note that, $\\mo_X(C)$ is locally free $\\mo_X$-module, hence flat. Therefore, tensoring this short exact sequence by $\\mo_X(C)$\n we get\n \\begin{equation}\\label{sh1}\n 0 \\to \\mo_X \\to \\mo_X(C) \\to \\N_{C|X} \\to 0 \n                 \\end{equation}                       \n                 is exact,\n where $\\N_{C|X}$ is the normal sheaf\n $\\Hc_X(\\mo_X(-C),i_*\\mo_C)$ which is isomorphic to the sheaf $i_*\\mo_C \\otimes_{\\mo_X} \\mo_X(C)$ (see \\cite[Ex. II.$5.1$(b)]{R1}). The \\emph{semi-regularity map}\n is the morphism \\[\\pi:H^1(\\N_{C|X}) \\to H^2(\\mo_X)\\]which arises from the long exact sequence associated to the short\n exact sequence (\\ref{sh1}). We say that $C$ is \\emph{semi-regular} if $\\pi$ is injective. \n\n\\end{para}\n\n\n\n\\begin{para}\n The Lefschetz hyperplane section theorem implies that $H^1(\\mo_X)=0$. Then, the long \n exact sequence associated to (\\ref{sh1}) contains the following segment:\n \\[0 \\to H^1(\\mo_X(C)) \\to H^1(\\N_{C|X}) \\xrightarrow{\\pi} H^2(\\mo_X).\\]\n So, $H^1(\\mo_X(C))=0$ is equivalent to $\\pi$ being injective, hence $C$ being semi-regular. We now prove a result that would help\n us determine when a curve is semi-regular.\n\\end{para}\n\n\\begin{lem}\\label{hf11}\nLet $C$ be a connected reduced curve and $d \\ge \\deg(C)+4$ then $h^1(\\mathcal{O}_X(C))=0$. In particular, $C$ is semi-regular.\n\\end{lem}\n\n\\begin{proof}\nSince $X$ is a hypersurface in $\\p3$ of degree $d$, $\\I_X \\cong \\mo_{\\p3}(-d)$.  Consider the short exact sequence:\n\\[0 \\to \\I_X \\to \\I_C \\to \\mo_X(-C) \\to 0.\\]\nTensoring this by $\\mo_{\\p3}(k)$, we get the following terms in the associated long exact sequence:\n\\[... \\to H^1(\\I_C(k)) \\to H^1(\\mo_X(-C)(k)) \\to H^2(\\I_X(k)) \\to ...\\]\nNow, $H^2(\\mo_{\\p3}(k-d))=0$ (see \\cite[Theorem $5.1$]{R1}) and $\\I_{C}$ is $\\deg(C)$-regular (see\n\\cite[Main Theorem]{gi1}). \nSo, $H^1(\\I_C(k))=0$ for $k \\ge \\deg(C)$. This implies $H^1(\\mo_X(-C)(k))=0$ for $k \\ge \\deg(C)$. \nBy Serre duality, $0=H^1(\\mo_X(-C)(d-4)) \\cong H^1(\\mo_X(C))$. So, $C$ is semi-regular.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\begin{para}\nLet $X$ be a surface and $C \\subset X$ be a curve.\n We now do a computation to show that for $d \\ge \\deg(C)+4$, $\\dim |C|=0$, where $|C|$ is the linear system of $C$ in $X$. \n\\end{para}\n\n\\begin{lem}\\label{a4e}\nLet $d \\ge 5$ and $C$ be an effective divisor on a smooth degree $d$ surface $X$ of the form $\\sum_i a_iC_i$, where $C_i$ are integral curves with $\\deg(C)+4 \\le d$.\nThen, $h^0(\\N_{C|X})=0.$ In particular, $\\dim |C|=0$, where $|C|$ is the linear system associated to $C$.\n\\end{lem}\n\n\\begin{proof}\nLet $C=\\sum_ia_iC_i$ with $C_i$ integral. Then, \\[\\deg((\\mathcal{O}_X(C)|_C \\otimes \\mo_C)|_{C_i})=a_iC_i^2+\\sum_{j \\not= i} a_jC_i.C_j.\\]\nDenote by $e_i:=\\deg(C_i)$. \nUsing the adjunction formula and the fact that $K_X \\cong \\mathcal{O}_X(d-4)$, we have that \n\\begin{eqnarray*}\n\\deg((\\mathcal{O}_X(C)|_C \\otimes \\mo_C)|_{C_i})&=&2a_i\\rho_a(C_i)-2a_i-(d-4)a_ie_i+\\sum_{j \\not= i}a_jC_i.C_j\\\\\n&\\le&a_i(e_i^2-(d-1)e_i)+\\sum_{j \\not=i}a_jC_iC_j\\\\\n&\\le&a_i(e_i^2-3e_i-e_i\\sum_ja_je_j)+\\sum_{j \\not=i}a_je_ie_j.\n\\end{eqnarray*}\nThe first inequality follows from the bound on the genus of a curve in $\\mathbb{P}^3$ in terms of its degree (see \\cite[Example $6.4.2$]{R1}).\nThe second inequality follows from the facts that $d \\ge \\deg(C)+4$ and $C_i.C_j \\le e_ie_j$. It then follows directly that $\\deg((\\mathcal{O}_X(C)|_C \\otimes \\mo_C)|_{C_i})<0$.\nThis implies that $h^0(C_i,(\\mathcal{O}_X(C)|_C \\otimes \\mc{O}_C)|_{C_i})=0$ for all $i$. So, \n$h^0(\\N_{C|X})=h^0(C,\\mathcal{O}_X(C)|_C \\otimes \\mo_C)=0.$\n\nSince $h^1(\\mathcal{O}_X)=0$ (by Lefschetz hyperplane section Theorem) and $h^0(\\mathcal{O}_X)=1$, using \nthe long exact sequence associated to the short exact sequence \n\\begin{equation}\\label{eq2}\n0 \\to \\mathcal{O}_X \\to \\mathcal{O}_X(C) \\to \\mathcal{O}_X(C)|_C \\otimes \\mo_C \\to 0\n\\end{equation}\nwe get that $h^0(\\mathcal{O}_X(C))=1$. Since $|C|=\\mb{P}(H^0(\\mo_X(C)))$, the lemma follows.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Flag Hilbert scheme and Hodge locus}\n\n\\begin{para}\n In this section we introduce the basic definitions of flag Hilbert schemes. See \\cite[\\S $4$]{S1} for further details. We then prove the main result of this section which relates Hodge locus to Hilbert schemes.\n\\end{para}\n\n\n\\begin{para}\nGiven an $m$-tuple of polynomials $\\mathcal{P}(t)=(P_1(t),P_2(t),...,P_m(t))$, we define the  contravariant functor, called the \\emph{Hilbert flag functor} \nrelative to $\\mathcal{P}(t)$,\\[FH_{\\mathcal{P}(t)}:(\\mbox{schemes}) \\to \\mbox{sets}\\]\n\\[S \\mapsto \\{(X_1,X_2,...,X_m)|X_1 \\subset X_2 \\subset ... \\subset \\mathbb{P}^3_S\\}\\]such that the Hilbert polynomial of $X_i$ is $P_i(t)$ and $X_i$ is an\n$S$-closed subscheme of $X_{i+1}$. We call such an $m$-\\emph{tuple a flag relative to} $\\mathcal{P}(t)$.\n\\end{para}\n\n\\begin{para}\n The functor $FH_{\\mathcal{P}(t)}$ is representable by a projective scheme, $H_{\\mathcal{P}(t)}$ which parametrizes all such flags relative to $\\mathcal{P}(t)$. We call this the \\emph{Hilbert flag scheme}.\n\\end{para}\n\n\n\n\n\\begin{thm}\\label{dim}\n Let $X$ be a surface, $C$ be a semi-regular curve in $X$ and $\\gamma \\in H^{1,1}(X,\\mb{Z})$ be the class of the curve $C$. For any irreducible component $L'$ of $\\overline{\\NL(\\gamma)}$\n (the closure is taken in the Zariski topology on $U_d$)\n there exists an irreducible component $H'$ of $H_{P,Q_d}$ containing the pair $(C,X)$ such that the associated reduced scheme $\\pr_2(H')_{\\red}$ \n coincides with $L'_{\\red}$, where $\\pr_2$ is \n the second projection map from $H_{P,Q_d}$ to $H_{Q_d}$. Furthermore, if $d \\ge \\deg(C)+4$ then such $H'$ is uniquely determined by $L'$.\n\\end{thm}\n\n\\begin{proof}\nThe first part of the theorem follows directly from \\cite[Theorem $7.1$]{b1}. \n\nFurthermore, Lemma \\ref{a4e} implies that $\\dim |C|=0$. So, given an irreducible component, say $L'$ of $\\overline{\\NL(\\gamma)}$ such that $X$ is a general\n  element, there exists an unique irreducible component $H'$ of $H_{P,Q_d}$ containing the pair $(C,X)$ such that $\\pr_2(H')_{\\red}$ coincides with $L'_{\\red}$. \n  This proves the rest of the theorem.\n  \n  \n  \n  \\end{proof}\n  \n  \n\n\n\n\n\n\n\n\n\n  \n  \n\n\n\n\n\n\n\\section{Variation of lattices}\n\\begin{para}\n In this section we give a formula to compute the dimension of an irreducible component of a Hodge locus (see Proposition \\ref{gh12}). \n This result will be particularly useful to prove the asymptotic case of a Griffiths-Harris conjecture, which we see in the next section.\n\\end{para}\n\n\n\\begin{para}\\label{pam6}\nLet $X$ be a surface of degree $d$. \nAn \\emph{augmented lattice } $\\Lambda_X$ \\emph{ on } $X$ \\emph{ of rank } $r$, is a rank $r$ $\\mb{Z}$-submodule $\\Lambda_X \\subset H^2(X,\\mb{Z})$ generated by the class of the very ample line bundle $H_X$ \n(as in \\ref{nl1})\nand cohomology classes of $r-1$ reduced curves, say $C_1,...,C_{r-1}$ such that\n$\\Lambda_X$ is saturated in the sense that for all $\\lambda \\in \\Lambda_X, c \\in \\mb{Q}$ if $c\\lambda \\in H^2(X,\\mb{Z})$ then $c\\lambda \\in \\Lambda_X$. For such $\\Lambda_X$, \nwe say that $C_i$ for $i=1,...,r-1$ \\emph{generate} $\\Lambda_X$.\nWe say that $\\Lambda_X$ is \\emph{prime} if $C_1,...,C_{r-1}$ are integral.\n\\end{para}\n\n\n\n\\begin{para}\\label{pam3}\nLet $\\Lambda_X$ be as in \\ref{pam6}. \nWe can define \\[ \\NL(\\Lambda_X):=\\{G \\in U | \\overline{\\gamma}_G=0, \\mbox{ for all } \\gamma \\in \\Lambda_X\\}.\\]\nFor a surface $X$ and reduced curve $C \\subset X$ such that the cohomology class $[C]$ of $C$ is not a $\\mb{Q}$-multiple of $c_1(H_X)$, denote by $\\Lambda_X^0$ the rank $2$ $\\mb{Z}$-module generated by $[C]$ and \n$c_1(H_X)$, where $c_1$ is the first Chern class map.\nSince a very ample line bundle remains of type $(1,1)$ is the family $\\mc{X}$, $\\NL(\\Lambda_X^0)$ coincides with $\\NL([C])$. More generally, for a rank $r$ augmented lattice $\\Lambda_X$ generated by $C_1,...,C_{r-1}$ we have,  $\\NL(\\Lambda_X)$ is isomorphic to the fiber product $\\NL([C_1]) \\times_{H_{Q_d}} ... \\times_{H_{Q_d}} \\NL([C_{r-1}])$.\n\\end{para}\n\n\\begin{para}\\label{pam2}\nLet $\\Lambda_X$ be as before of rank $2$, generated by a reduced curve, say $C$. \nLet $P$ be the Hilbert polynomial of $C$. Assume $d \\ge \\deg(C)+4$. Using Theorem \\ref{dim} we can conclude that for general $X' \\in \\NL(\\Lambda_X)$ there exists a curve $C' \\subset X'$ such that \n$\\overline{\\NL([C'])}$ is an irreducible component of $\\overline{\\NL(\\Lambda_X)}$ and $C'$ deforms to $C$, i.e., $C'$ has the same Hilbert polynomial $P$. Denote by $\\Lambda_{X'}$ the augmented lattice on $X'$ of \nrank $2$ generated by $C'$.\nTheorem \\ref{dim} again implies that there exists an unique irreducible component, denoted $H_{\\Lambda_{X'}}$, of $H_{P,Q_d}$ containing the pair $(C',X')$ such that $\\pr_2(H_{\\Lambda_{X'}})$ is isomorphic to $\\overline{\\NL(\\Lambda_{X'})}$.\nDenote by $L_{\\Lambda_{X'}}:=\\pr_1(H_{\\Lambda_{X'}})$. From now on, we will always assume that $\\overline{\\NL(\\Lambda_X)}$ is irreducible, which is equivalent to $X$ \nbeing general in $\\NL(\\Lambda_X)$, in particular,\naway from the points of intersection of any two irreducible components of $\\overline{\\NL(\\Lambda_X)}$. \n\\end{para}\n\n\\begin{para}\nSuppose now that $\\Lambda_X$ is of rank $r$ generated by $C_1,...,C_{r-1}$. Let $P_1,...,P_{r-1}$ be the Hilbert polynomials of $C_1,...,C_{r-1}$, respectively. Consider the natural morphism  \n \\[p:H_{P_1,Q_d} \\times_{H_{Q_d}} ... \\times_{H_{Q_d}} H_{P_{r-1},Q_d} \\to H_{Q_d}.\\] Assume $d \\ge \\sum_{i=1}^{r-1}\\deg(C_i)+4$. Using Theorem \\ref{dim}, we can conclude that for every irreducible component $L'$\n of $\\overline{\\NL(\\Lambda_X)}$ there exists an unique irreducible component, say $H$ of $H_{P_1,Q_d} \\times_{H_{Q_d}} ... \\times_{H_{Q_d}} H_{P_{r-1},Q_d}$ containing $(C_1,X) \\times ... \\times (C_{r-1},X)$ \n such that $p(H)$ coincides with $L'$. Similarly as in \\ref{pam2}, by taking $X$ general in $\\NL(\\Lambda_X)$ we can ensure that $\\overline{\\NL(\\Lambda_X)}$ is irreducible. \n Denote by $H_{\\Lambda_X}$ the irreducible component of $H_{P_1,Q_d} \\times_{H_{Q_d}} ... \\times_{H_{Q_d}} H_{P_r,Q_d}$ such that $p(H_{\\Lambda_X})$ coincides with $\\overline{\\NL(\\Lambda_X)}$. \n Denote by $L_{\\Lambda_X}:=\\pr(H_{\\Lambda_X})$, where $\\pr$ is  the natural projection map from $H_{P_1,Q_d} \\times_{H_{Q_d}} ... \\times_{H_{Q_d}} H_{P_{r-1},Q_d}$ to $H_{P_1} \\times ... \\times H_{P_{r-1}}$.\n\\end{para}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{prop}\\label{gh12}\nLet $r \\ge 3$, $X$ be a surface of degree $d$ and $\\Lambda_X$ be an augmented lattice of rank $r+1$ generated by $r$ reduced curves $C_1,...,C_r$. Assume that $\\sum_{i=1}^r \\deg(C_i)+4 \\le d$.\nThen, the dimension of $\\ov{\\NL(\\Lambda_X)}$ is given by the following formula: \\[\\codim \\ov{\\NL(\\Lambda_X)}=\\codim I_d(C)-\\dim L_{\\Lambda_X}\\] where $C=C_1' \\bigcup ... \\bigcup C_r'$ for a  general $r$-tuple $(C_1',...,C_r')$ in $L_{\\Lambda_X}$\nand $I_d(C)$ is the degree $d$ graded piece in the ideal $I(C)$ of $C$.\n\\end{prop}\n\n\\begin{proof}\nConsider the diagram,\n\\[\\begin{diagram}\nH_{\\Lambda_X}&\\rTo^{\\pr_1}&L_{\\Lambda_X}\\\\\n\\dTo^{\\pr_2}& \\\\\n\\ov{\\NL(\\Lambda_X)}\\\\\n\\end{diagram}\\]\nDenote by $P_i$ the Hilbert polynomial of curves $C_i$, respectively. Recall, $L_{\\Lambda_X}$ is contained in $H_{P_1} \\times ... \\times H_{P_r}$. \nFor an $r$-tuple $(C_1,...,C_r) \\in L_{\\Lambda_X}$, the fiber of $\\pr_1$ parametrizes the space of smooth degree $d$ surfaces containing $C=C_1 \\bigcup ... \\bigcup C_r$, which is an open subscheme in $\\mb{P}(I_d(C))$.\nSince $I_d(C)$ is irreducible, the dimension of the generic fiber of $\\pr_1$ is equal to $\\dim I_d(C)-1$, where $(C_1,...,C_r) \\in L_{\\Lambda_X}$ is a general element.\nThe fiber of $\\pr_2$ over $\\pr_2((C_1,...,C_r,X))$ is isomorphic to $|C_1|\\times ... \\times |C_r|$. But, Lemma \\ref{a4e} implies $\\dim |C_i|=0$ for $i=1,...,r$.\nSo, the dimension of the generic fiber of $\\pr_2$ is zero.\nThen, \\[\\dim H_{\\Lambda_X}=\\dim L_{\\Lambda_X}+\\dim I_d(C)-1=\\dim \\NL(\\Lambda_X).\\]\n\\[\\mbox{So, } \\codim \\overline{\\NL(\\Lambda_X)}=\\dim \\mb{P}(H^0(\\mo_{\\p3}(d)))-\\dim \\overline{\\NL(\\Lambda_X)}=h^0(\\mo_{\\p3}(d))-\\dim I_d(C)-\\dim L_{\\Lambda_X}.\\]\nThis finishes the proof of the proposition.\n\\end{proof}\n\n\n\n\n\\section{A Griffiths-Harris conjecture}\n\n\n\n\n\\begin{para}\nWe now come to the final section of the article, where we prove an asymptotic case of a Griffiths-Harris conjecture. Recall,\na \\emph{Griffiths-Harris conjecture} states in \\cite{GH} that: \n\\begin{center}\nFor $3 \\le r \\le d$, the codimension of an irreducible component of $\\NL_{r,d}$ is at least equal to \\[(r-1)(d-3)-\\binom{r-3}{2}.\\]\nFurthermore, there exists a component of $\\NL_{r,d}$ of this codimension parametrizing the space of surfaces \ncontaining $r-1$ lines on the same plane. \n\\end{center}\n\\end{para}\n\n\\begin{note}\nWe will denote by $\\N_d(r)$ the number, \\[(d-3)(r-1)-\\binom{r-3}{2}.\\]We now recall a result in Noether-Lefschetz locus due to Otwinowska which will help us characterize the irreducible components of $\\NL_{r,d}$ with codimension\nless than or equal to $\\N_d(r)$.\n\\end{note}\n\n\n\\begin{thm}[{\\cite[Theorem $1$]{ot}}]\\label{gh22}\n Let $\\gamma$ be an augmented lattice of rank $2$ on a degree $d$ surface. There exists $C \\in \\mathbb{R}_+^*$ depending only on $r$ such that for $d \\ge C(r-1)^8$ if $\\codim \\NL(\\gamma) \\le (r-1)d$ \n then $\\gamma_{\\prim}=\\sum_{i=1}^ta_i[C_i]_{\\prim}$, where $a_i \\in \\mathbb{Q}^*$, $C_i$ are reduced curves and $\\deg(C_i) \\le (r-1)$ for\n $i=1,...,t$ for some positive integer $t$.\n\\end{thm}\n\n\\begin{para}\nThroughout this section we denote by $r$ an integer greater than or equal to $3$ and for a fixed $r$, denote by $d$, an integer as mentioned in Theorem \\ref{gh22}. We will \\emph{assume} \nthat $d$ is at least $r^3$ which will be used only in a computation in Lemma \\ref{gh4}. The other results do not have any restriction on $d$ in terms of $r$.\n\\end{para}\n\n\\begin{prop}\\label{c3}\nLet $L$ be an irreducible component of $\\NL_{r,d}$. Then $L$ is locally homeomorphic to $\\NL(\\Lambda)$ for some prime augmented lattice $\\Lambda$ of \nrank at least $r$ on a surface $X \\in L$, general.\n\\end{prop}\n\n\\begin{proof}\nLet $L \\subset \\NL_{r,d}$ be an irreducible component. Let $X$ be a general element in $L$. This implies that for the Picard lattice $\\Lambda:=\\NS(X)$,  $\\NL(\\Lambda)_{\\red}$ is an open subscheme of $L_{\\red}$.\nWe can assume that $\\Lambda$ is a prime lattice. Since $X$ is an element in $\\NL_{r,d}$, the rank of $\\Lambda$ is greater than or equal to $r$.\n\\end{proof}\n\n\n\n\\begin{prop}\\label{gh14}\nIf $\\Lambda$ is an augmented prime lattice of rank $t$ on some degree $d$ surface and $\\codim \\overline{\\NL(\\Lambda)} \\le (r-1)d$. \nThen there exists a prime lattice $\\Lambda'$ of rank greater than or equal to $t$ generated by classes of curves of degree less than or equal to $r-1$,\nsuch that $C_i$ deforms along $\\NL(\\Lambda')$ and\n$\\overline{\\NL(\\Lambda)}_{\\red}=\\overline{\\NL(\\Lambda')}_{\\red}$.\n\\end{prop}\n\n\\begin{proof}\n\n\n\n\n \nLet $X \\in \\NL(\\Lambda)$.\nThere exists a maximal lattice $\\Lambda' \\subset H^2(X,\\mb{Z})$ such that $\\Lambda'$ remains of type $(1,1)$ in $\\NL(\\Lambda)$ i.e., $\\NL(\\Lambda)_{\\red}=\\NL(\\Lambda')_{\\red}$. \nNow, there exists \na surface $X' \\in \\NL(\\Lambda')$ such that the N\\'{e}ron-Severi group $\\mr{NS}(X')$ is the translate (under deformation from $X$ to $X'$) of \n$\\Lambda'$ in $H^2(X',\\mb{Z})$ which we again denote by $\\Lambda'$ for convinience of notation. Then Theorem \\ref{gh22} implies that  any $\\gamma \\in \\Lambda'$ is of  the form $\\sum_i a_i[C_i]+bH_X$ with $\\deg(C_i) \\le r-1$.\nSo, $\\Lambda'$ can be generated by classes of curves of degree at most $r-1$ and the class of the very ample line bundle $H_X$. \nNow, the class of $[C_i]$ remains of type $(1,1)$ along $\\NL(\\Lambda')$. From Lemma \\ref{hf11} it follows that $C_i$ is semi-regular. Then, \\cite[Theorem $7.1$]{b1}\nimplies that the class of $[C_i]$ remains effective along $\\NL(\\Lambda')$.\nThis proves the proposition.\n\\end{proof}\n\n\\begin{para}\n We now recall a result due to Eisenbud and Harris which we use in the next lemma. Let $P$ be a Hilbert polynomial of a curve in $\\p3$ of degree $e$ and $L$ be an irreducible component of $H_P$.\n The corollary after \\cite[Theorem $1$]{eis} tells us that,\n \\begin{thm}[\\cite{eis}]\\label{ei1}\n  For $e>1$, the dimension of $L$ is less than or equal to $3+e(e+3)\/2$.\n   \\end{thm}\n\\end{para}\n\n\n\\begin{lem}\\label{gh4}\nLet $\\Lambda$ be a prime augmented lattice of rank $t+1$ on a degree $d$ surface, generated by irreducible curves $C_i$ for $i=1,...,t$ for some positive integer $t$\nand $\\deg(C_i) \\le r-1$. Suppose $\\codim \\NL(\\Lambda)\\le (r-1)d$. Then, $\\sum_{i=1}^t \\deg(C_i) \\le (r-1)$.\n\\end{lem}\n\n\\begin{proof}\nWe prove this by induction on $t$. This is trivially true for $t=1$.\nSuppose this is true for all $t \\le m$. \n\nAssume this is not true for $t=m+1$. In other words, there exists a prime lattice $\\Lambda$ minimally generated by $m+1$ curves such that \n$\\sum_i \\deg(C_i)>r-1$. \nThis implies (after rearranging the indices if necessary) there exists an integer $0 <t' \\le m+1$ such that $C_1,...,C_{t'}$ satisfies $\\sum_{i=1}^{t'} \\deg(C_i)>(r-1)$ and $\\sum_{i=1}^{t'-1} \\deg(C_i) \\le (r-1)$.\nThen, $\\sum_{i=1}^{t'} \\deg(C_i) \\le 2(r-1)$.\nDenote by $P$ the Hilbert polynomial of the curve $C_1 \\bigcup ... \\bigcup C_{t'}$.\nWe replace $e$ by $2(r-1)$ in Theorem \\ref{ei1} and conclude that the dimension of the Hilbert scheme $H_P$ \nis less than or equal to $3+(r-1)(2r+1)$. Using Proposition \\ref{gh12}, the codimension \nof $\\NL([C_1+...+C_{t'}])$ is greater than or equal to $\\codim I_d(C_1 \\bigcup ... \\bigcup C_{t'})-\\dim H_P$. Since $r-1<\\deg(C_1+...+C_{t'})\\le 2(r-1)$ and $d \\ge r^3$, we get the following inequality using the upper bound on \nthe arithmetic genus of a curve of degree less than or equal to $2(r-1)$:\n\\begin{eqnarray*}\n \\codim \\NL([C_1+...+C_{t'}]) & \\ge &\\codim I_d(C_1+...C_{t'})-\\dim H_P\\\\\n &\\ge& (rd-(2r-3)(2r-4)\/2+1)-(3+(r-1)(2r+1))\\\\\n &=&(r-1)d+(d-(2r-3)(2r-4)\/2-3-(r-1)(2r+1)+1)\\\\\n &>&(r-1)d\n\\end{eqnarray*}\ncontradicting the assumption.\n\\end{proof}\n\n\n\n\n\\begin{prop}\\label{gh13}\nLet $\\Lambda$ be an augmented lattice of rank $r$ contained in a degree $d$ surface, generated by $l_i$ for $i=1,...,r-1$, where $l_i$ are lines for \nall $i$. Suppose $r \\ge 3$. Then, $\\codim \\NL(\\Lambda) \\ge \\N_d(r)$. Furthermore, if $l_i$ are on the same plane we have\nan equality.\n\\end{prop}\n\n\\begin{proof}\nWe prove this by induction.\n Using Proposition \\ref{gh12}, \\[\\codim \\NL(\\Lambda)=\\codim I_d(\\bigcup_{i=1}^{r-1} l_i)-\\dim L_{\\Lambda}.\\]\n If $r=3$, $\\codim \\NL(\\Lambda)=2d-6=\\N_d(3)$.\n Assume the result holds true for all $r \\le m$ for some integer $m \\ge 4$. We now prove for $r=m+1$.\nDenote by $\\Lambda'$ the lattice generated by $l_i$ for $i=1,...,r-2$ and by $C$ the curve $\\bigcup_{i=1}^{r-2} l_i$. \nLet $t:=\\dim L_{\\Lambda}-\\dim L_{\\Lambda'}$. Note that, $t \\le 4$. \n\nNote that $l_{r-1}.C \\le r-2$.\nDenote by $\\epsilon:=r-2-l_{r-1}.C$. Comparing with the list of values of $t$, we see \n\\begin{enumerate}\n \\item If $\\epsilon=0$ then $l_{r-1}.C=r-2$ and $t \\le 2$. In particular, for a fixed curve $C$ there is a $1$-$1$ correspondence between the \n set of choices of $l_{r-1}$ intersecting $C$ in $r-2$ points and the set of planes $P$ interesting $C$ at $r-2$ distinct collinear points. If for a generic choice of $P$, $P \\bigcap C$ are $r-2$ distinct collinear points\n then all lines in $C$ should lie on the same plane. In that case, $l_{r-1}$ intersect $C$ at $r-2$ points if and only if $l_{r-1}$ is on the same plane as $C$, hence $t=2$ (dimension of the space of lines in $\\mb{P}^2$). \n If this is not the case i.e., \n a generic choice of $P$ does not intersect $C$ in $r-2$ distinct collinear points, then $t \\le 2$.\n\\item $\\epsilon=r-2$ and $t=4$ if $l_{r-1}$ does not intersect $C$.\n\\item $0 < \\epsilon <r-2$ and $t \\le 3$ otherwise.\n\\end{enumerate}\n\nNow, \n\\begin{eqnarray*}\n \\codim \\NL(\\Lambda)&=&\\codim I_d(C \\bigcup l_{r-1})-\\dim L_\\Lambda\\\\\n &=&(\\codim I_d(C)+\\codim I_d(l_{r-1})-l_{r-1}.C)-\\dim L_{\\Lambda}\\\\\n &=&(\\codim I_d(C)-\\dim L_{\\Lambda'})+(\\codim I_d(l_{r-1})-t)-(r-2-\\epsilon).\n\\end{eqnarray*}\nWriting $t=4-(4-t)$ and using the induction step, we see that the right hand side of this equation is greater than\nor equal to \n\\begin{eqnarray*}\n ((r-2)(d-3)-\\binom{r-4}{2})+(d-3)-(r-2-\\epsilon)+(4-t)\\\\\n =((r-2)(d-3)-\\binom{r-3}{2})+\\epsilon-t+2\n\\end{eqnarray*}\nNote that $\\epsilon-t+2 \\ge 0$ from the three cases considered above. Substituting this inequality gives us\n$\\codim \\NL(\\Lambda) \\ge \\N_d(r)$. \n \\end{proof}\n\n\\begin{thm}\\label{gh6}\nLet $L$ be an irreducible component of $\\NL_{r,d}$. Then, $\\codim L \\ge (r-1)(d-3)-\\binom{r-3}{2}$.\nFurthermore, there exists a component $L$ of $\\NL_{r,d}$ of this codimension parametrizing surfaces containing $r-1$ coplanar lines.\n\\end{thm}\n\n\\begin{proof}\nIt suffices to prove $L$ is locally of the form $\\NL(\\Lambda)$ for $\\Lambda$ a rank $r-1$ prime lattice generated by $[l_i]$\nfor $i=1,...,r$, where $l_i$ are lines on the same plane. Using Proposition \\ref{c3}, $L$ is of the form $\\NL(\\Lambda)$ for a rank $r-1$ prime\nlattice $\\Lambda$. Using Proposition \\ref{gh14} we can assume that there exists a prime lattice $\\Lambda'$ of rank $t$ \ngreater than or equal to $r-1$ generated by classes of curves, say $C_i'$ for $i=1,...,t$ of degree less than or equal to $r-1$ such that \n$\\NL(\\Lambda)=\\NL(\\Lambda')$. Lemma \\ref{gh4} implies that $t=r-1$ and $\\deg(C_i')=1$ for all $i=1,...,r-1$.\nFinally, the theorem follows from Proposition \\ref{gh13}.\n\\end{proof}\n\n\n\n\\bibliographystyle{alpha}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the present paper we explore large photometric data sets of four yellow hypergiants (YHGs) \\object{HR\\,8752\\,=\\,V509\\,Cas}, \\object{HR\\,5171A\\,=\\,V766\\,Cen,} and \\object{$\\rho$~Cas\\,=\\,HD\\,224014} (the Big Three), and \\object{HD\\,179821\\,=\\,V1427 Aql}, the class of which is still undetermined (Arkhipova et al. 2009; Oudmaijer et al. 2009; Le Coroller et al. 2003; Ferguson \\& Ueta 2010; Sahin et al. 2016). This latter could be a YHG or a red supergiant (RSG), but also a post-asymptotic giant branch(AGB) star, thus a much lower-mass object and has been incorporated into this study due to its spectroscopic and photometric similarities with both classes. The three types of objects, YHGs, RSGs and post-AGBs, are almost entirely convective.\n\nLate-type supergiants showing the H$\\alpha$ line with one or more broad emission components and exceptional broad absorption lines in their spectra are called hypergiants Ia$^{+}$. The approximate ranges for the M$_{\\rm bol}$\\,=\\,-8.7\\,--\\,-9.6, for the luminosity log\\,L\/L$_{\\rm \\odot}$\\,=\\,5.3--5.7 and for the temperature T$_{\\rm eff}$\\,=\\,4000\\,K--6000\\,K (excluding eruptive episodes). Typical for these stars are their very extended atmospheres, surface gravities around zero, high mass-loss rates, and strongly developed large-scale photospheric and atmospheric motion fields (de Jager 1980, 1998). The $\\kappa$-mechanism of the two partially ionized He zones is likely responsible for the pulsations with a quasi-period of a few hundred days (e.g. Fadeyev 2011).\nThese stars are also in the state of gravitational contraction of their He core (Meynet et al. 1994; Stothers \\& Chin 1996).\n\nOn a timescale of one to a few decades, a pulsation develops into an atmospheric eruptive episode, lasting about one to two years, reaching a high temperature (6500\\,K--7500\\,K; we note that this is not an evolutionary temperature rise). Lobel (2001) presented non-LTE calculations, showing that in this temperature range, cool supergiants become dynamically unstable due to the decrease of the first adiabatic index stability-integral $\\langle$$\\Gamma_{1}$$\\rangle$ over a major fraction of the extended atmosphere as a result of partial, thermal and photo-ionization of hydrogen. At these high T$_{\\rm eff}$ large fractions of partially ionized H gas\n(T\\,gas\\,$\\simeq$\\,8000\\,K) can synchronously recombine under pulsation decompression and drive the fast expansion of the entire atmosphere in an outburst event with strong global cooling. An enormous amount of energy is released which blows away part of the upper atmospheric layers.\nSubsequently, a deep light minimum follows with T$\\sim$\\,4300\\,K and the spectrum shows TiO absorption bands. The mass-loss rate as recorded during the  eruption of $\\rho$\\,Cas in the year 2000 amounted to $\\sim$~3.10$^{-2}$\\,M$_{\\sun}$yr$^{-1}$ (Lobel et al. 2003). This amount is of the same order as that predicted by hydrodynamical models by Tuchman et al. (1978) and Stothers (1999).\n\nFigure\\,1 depicts a schematic HR diagram.\nA cool portion of the HR diagram appeared to be almost void of stars (Humphreys 1978). Part of this void was theoretically studied and defined by de Jager (1998) and observationally checked by Nieuwenhuijzen \\& de Jager (2000). Very relevant for this study was also the comparison of the evolutionary tracks of HR\\,8752 and other YHGs with the models of Meynet et al. (1994) by Nieuwenhuijzen et al. (2012) in their Fig.\\,1 and Sect.\\,1.2. In this latter analysis,  IRC\\,+10240 also appeared to be a YHG moving on an evolutionary track (Oudmaijer (1998).\n\nThis area was divided into two voids: the Yellow Void and the Yellow-Blue Void. These represent critical phases in the evolution of YHGs, bounded by the three semi-vertical curves. The curve at lowest T$_{\\rm eff}$\\,$\\sim$~5200\\,K is at photospheric values where H is close to beginning its ionization. The next curve at $\\sim$\\,7500\\,K (dashed) is at photospheric values where H is close to being completely ionised and the curve at the highest T$_{\\rm eff}$ between 8900\\,K and 12500\\,K marks the parameters where neutral He is close to being fully ionized.\n\nThe extreme photospheric values of the four YHGs studied in this paper are given in Table\\,A:Data summary of the Appendix. It should be noted that with the exception of HR\\,8752, no Hipparcos or Gaia Data Processing and Analysis Consortium (DPAC) parallaxes were used for the distances. Together with other experts in this field we advise against using these latter for very big stars and stars brighter than six, as then the reliability of parallaxes is very uncertain (van Leeuwen, priv. comm. 2019).\n\nTo prevent crowding in Fig.\\,1, only one or two selected red or blue evolutionary loops (hereafter referred to as RL and BL, respectively) per YHG are schematically represented by lines (or dotted curves), and their observed evolutionary directions by an arrow.\n\nFor HR\\,8752: a RL and a BL (based on Nieuwenhuijzen et al. 2012) are connected by the curved dotted line (during which the 1973 eruption happened) to indicate the evolutionary direction (not the true track).\n\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics{arnout.pdf}}\n  \\caption{A schematic HR diagram: on the right and left of the vertical dashed line the Yellow Void and the Yellow-Blue Void, respectively (data in Table\\,A1:Data summary of the Appendix). The diagram shows the extreme physical and photometric values of one or two (HR\\,8752) selected RL or BL evolutionary tracks of the program stars (arrows). Most of the extreme values are accidentally concentrated near the vertical curve on the right, the curve where H is close to beginning their ionization.}\n  \\label{Fig.1}\n   \\end{figure}\n\nHow long this zigzag behaviour lasts is unknown; it is meant to lose enough mass (mainly during the eruptive episodes) in order to evolve further to the blue to eventually become an S\\,Dor variable (also referred to as a luminous blue variable (LBV)), or a Wolf-Rayet star (WR-star) (Oudmaijer et al. 2009; Oudmaijer \\& de Wit 2013).\n\nCritical comments by Sterken (2014, priv.comm.) on the supposition that HR\\,5171A is a contact binary on account of a double-waved light curve constructed by Chesneau et al. (2014) contributed to our growing realization at the time (2014) that little was known about the photometric properties of the light curves of individual quasi-periodic pulsations and long-term light variations (LTVs) of YHGs.\n\nTherefore, our focus was to probe the large photometric databases, comprising 50 to 120\\,yr of observations of the four selected YHGs to the bottom, otherwise pivotal developments will be absent. The most important aims of this study are\nto find out why spectroscopic temperatures are always higher than photometric ones; to decipher\nwhether or not in addition to HR\\,8752 (Nieuwenhuijzen et al. 2012), the three other selected YHGs RL and BL evolutions on timescales of decades, interrupted by eruptive episodes;\nto find out whether or not the observed LTVs represent these evolutionary tracks; and to find out whether the light variations of  HR\\,5171A are due to pulsations alone, and why it reddened stubbornly from 1960 to 1981.\nWe also aim to find an explanation for the variable trends of quasi-periods.\n\nSections\\,1 and 2 are devoted to the $\\sim$\\,1\\,yr stellar instabilities and the stellar properties.\nSection\\,3 is devoted to the connection between LTVs and the evolutionary loops of YHGs on the HR diagram and the variable trends of quasi-periods. Section\\,4 is devoted to a number of theoretical and speculative explanations for some observational highlights.\nSection\\,5 summarises the most important novel findings.\n\n\\section{Miscellaneous photometric and physical details and pulsation properties. }\n\n\\subsection{Databases and published extinctions.}\n\nBased on decades up to more than a century of visual, mono-, and multi-colour photometry, it was obvious that the four selected objects show more or less the same pattern of 'semi-regularity' of pulsations, and appear coherent. The quasi-periods are a few hundred days in length and the pulsations are superimposed on the LTVs;\nwe refer to Appendix\\,B:\\,2.1. for references and descriptions.\n\nInterstellar extinctions and reddening are required for our calculations of stellar properties: these are summarised in Table\\,1.\n\n\\begin{table}\n\\caption{Adopted interstellar extinctions, reddening, and the extinction laws R for the four objects used in this paper. The reddening values, dependant on the extinction law R, in the Johnson $UBVRI$ were retrieved from the tables of Steenman \\& Th\\'e (1989).}\n\\label{table:1}\n\\centering\n\\begin{tabular}{c|c|c|c|c}\n\\hline\\hline\n & HR\\,8752 & $\\rho$\\,Cas & HR\\,5171A & HD\\,179821 \\\\\n \\hline\n A$_{v}$ & 3.08$^{1}$ & 1.44$^{2}$ & 4.32$^{3}$ & 2.24$^{3}$ \\\\\n R &       4.4$^{1}$ & 3.2 $^{3}$ & 3.6$^{5}$ & 3.2$^{3}$ \\\\\n E$(B-V)$ & 0.70$^{1}$ & 0.45$^{4}$ & 1.2$^{3}$ & 0.7$^{6}$ \\\\\n E$(U-B)$ & 0.67     & 0.44              \\\\\nE$(V-R)$ &  0.66         & 0.40     \\\\\nE$(V-I)$  &  1.31           & 0.73     \\\\\nE$(R-I)$  &  0.65          & 0.34     \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{1\\,=\\,Nieuwenhuijzen et al. (2012); 2\\,=\\,Joshi \\& Rautela (1978); 3\\,=\\,This paper, E$(B-V)$\\,=\\,0.4 for ISE and 0.8 for CSE; 4\\,=\\,Lobel et al. (2003); 5\\,=\\,van Genderen et al. (2015); 6\\,=\\,Arkhipova et al. (2009)}\n\\end{table}\n\n\\subsection{Pulsation properties. }\n\nFigures\\,2 and 3 show two typical time-series for the YHG cadence of pulsations. The first represents the very accurate $\\rho$\\,Cas photometry in $V$ and $B$ by Henry (1995, 1999) for the time interval 2003--2017 relative to the comparison star HD\\,223173\\,=\\,HR\\,9010 ($\\triangle$ signifies variable minus comparison star). Henry's observations concern the T2 $VRI$ (1986--2001) and the T3 $BV$ (2003--2018) Automatic Telescope Projects (ATP). For the relative magnitudes we used for the comparison star: (Johnson system) $V$\\,=\\,5.513, $B$\\,=\\,7.160, $R$\\,=\\,4.220, $I$\\,=\\,3.390. See stub-tables Tables\\,M.1. and M.2. in the Appendix for Henry's complete tables in electronic form available at the CDS.\nThe mean lines represent the LTVs. Between 2007 and 2011 (JD\\,24\\,54168--JD\\,24\\,55819) Klochkova et al. (2014) derived ten spectral temperatures (T(Sp)) of between 5777\\,K and 6744\\,K, thus showing a range of $\\sim$\\,1000\\,K.\n\nThe two panels of Fig.\\,3 show the cadence pattern of HR\\,5171A (1989--1995 and 2009--2016, respectively), which is not significantly different from those of $\\rho$\\,Cas in Fig.\\,2, of HR\\,8752 (not depicted), and of HD\\,179821 (e.g. Arkhipova et al. 2009).\nBoth panels of Fig.\\,3 also show a significant declining visual brightness of the LTV.\n\nIn the upper panel of Fig.\\,3 we matched professional photometry in $V$ based on several sources: van Genderen (1992, hereafter called Paper\\,I); Chesneau et al. 2014) and visual observations by the American Association of Variable Star Observers (AAVSO) represented by the red curves which are based on the smoothed black dots. AAVSO data were shifted 0\\fm4 upwards, the Hipparcos $H_p$ magnitudes shifted 0\\fm1 downwards, and the $y$ ($uvby$) of the Str\\\"omgren system were shifted 0\\fm1 upwards in order to match the $V$ of the Walraven $VBLUW$ system transformed into $V$ of the $UBV$ Johnson system (Paper I).\n\n\\begin{figure*}\n \\sidecaption\n  \\includegraphics[width=12cm]{img308.pdf}\n   \\caption{The 2003-2017 time-series in $B$ and $V$ ($UBV$) of $\\rho$\\,Cas relative to the comparison star by the T3 APT project (Henry 1995, 1999). Small dots on the right in $V$ and $B$ are observations made by one of the present authors (EJvB; see Appendix B: 2.1.). The mean curves sketched through the median magnitudes of the pulsations are the LTVs, in $V$ almost constant, in $B$ clearly present. Some numbers in the $B$ light curve represent pulsations quoted in the text. The sequence contains one new but weak eruption: 2013 (labelled no.8 in the $B$-panel and recognizable by its deep minimum). The pulsation in 2004, labelled no.\\,7, has a relatively deep minimum, but is not an eruption episode. Spectral temperatures were derived by Klochkova et al. (2014) between JD\\,24\\,54168 and JD\\,24\\,55819.}\n  \\label{Fig2}\n  \\end{figure*}\n\nNo shift of the magnitude scale for the AAVSO data has been applied in the lower panel showing the difference with the six photo-electric $V$ magnitudes represented by plus signs (these $BV$ Johnson and $RI$ Cousins data are listed in Table\\,I.1. of the Appendix). The systematic difference between the two observation techniques is common knowledge (Bailey 1978): the visual data are generally fainter by 0\\fm2-0\\fm4 than the photo-electric data (in the upper panel we applied a correction of 0\\fm4).\nThe vertical line in the lower panel near JD\\,24\\,56755 represent an observation made with the aid of the Astronomical Multi-BEam combineR (AMBER) instrument of the VLTI instrument at the ESO by Wittkowski et al. (2017a), indicating that T$_{\\rm eff}$\\,=\\,4290\\,K\\,$\\pm$\\,760\\,K. We estimate that the brightness was $V$\\,$\\sim$\\,6.4.\n\nOur definition of the quasi-period, or duration $\\triangle$\\,D in days, is through timing measurements of light maxima. The amplitudes of the pulsations of the four objects vary between 0\\fm2 and $\\sim$\\,0\\fm5, and are variable from cycle to cycle.\n\nEruptions are striking, among other things, by the relatively large depth of the minimum ($\\sim$\\,1$^{\\rm m}$) after the eruption maximum, called max1, of which the amplitude is sometimes slightly higher than for normal pulsation maxima. The duration of the entire eruption episode is about twice the duration of an ordinary pulsation. Because of such markers we identified one new eruption in Fig.\\,2 ($B$ panel): in 2013 (the maximum named no.\\,8). This eruption has been independently confirmed by spectroscopic observations (Aret et al. 2016). However, it was of a much weaker caliber than those of for example 1986 and 2000.\nThe pulsation in 2004, of which the maximum is labelled no.\\,7 in Fig.\\,2, showed more or less the same markers, but its light curve was even weaker than that of 2013, and it is definitely not an eruption (no spectroscopic indications).\n\n\\begin{figure*}\n  \\sidecaption\n   \\includegraphics[width=12cm]{img019.pdf}\n \\caption{Upper panel:  1989--1995 pulsations of HR\\,5171A in the visual (the averages of the AAVSO data shifted upward by 0\\fm4 (black dots)\n from which smoothed averages were derived: the red dots). Other data points are added (e.g. Chesneau et al. 2014): the $V$ (Johnson system, transformed from the $V$ of the $VBLUW$ Walraven\n    system, light green squares), the Hipparcos $Hp$ magnitudes shifted by 0\\fm1 downwards (dark\n   blue squares), and the $y$ ($uvby$) Str\\\"omgren system, shifted by 0\\fm1 upwards (light blue, cyan squares). In this way all data fit the $V$ (Johnson system)\n   magnitude scale satisfactorily. The deepest minimum on the right ($\\sim$\\,JD 244\\,9750) is called the Jones-Williams minimum.\n   The lower panel (we highlight the small scale difference between both JD axes) shows the 2010--2016 pulsations (based on AAVSO data). The last pulsation on the right with the minimum at $\\sim$\\,JD\\,24\\,57330 (early October 2015) is called the Blown-minimum. Six photo-electric\n $BV$ data (Johnson system) and $RI$ (Cousins system) observed by one of the present authors (GDS) are only for $V$ represented by a `+', and are listed in\\,Table\\,I.1. of the Appendix together with four more recent observations. No magnitude scale shift has been applied. The vertical line piece at JD\\,24\\,56755 (2014) marks the date of an AMBER\/VLTI observation by Wittkovski et al. (2017a); see text. }\n \\label{Fig3}\n \\end{figure*}\n\nThere is some ambiguity about the two high peaks near JD\\,24\\,56500 in the lower panel of Fig.\\,3, but because of the large scatter of the observations we considered them tentatively as a single peak. Errors for visual magnitudes of very red objects like HR\\,5171A, are larger than those for less-red stars because of the Purkinje effect.\nA quasi-period search in the literature is summarized in Appendix\\,C:\\,2.2.\n\nAfter inspection of the large body of recorded photometric time-series of $\\rho$\\,Cas, HR\\,5171A, HR\\,8752, and HD\\,179821, our conclusion is that YHGs are subject to a coherent sequence of pulsations, but that each pulsation is unique. These pulsations are quasi-periodic. Additionally, the quasi-periodicity is uninterrupted without any cessation of pulsations (including the eruptive pulsation episodes lasting twice as long). Only light curves of great precision reveal bumps, plateaus, and depressions lasting for weeks to months.\n\nThis incessant cyclic behaviour without any gaps in the sequence, which otherwise would signal one or more missing cycles (e.g. in case of one small-scale non-radial pulsation only happening accidentally on the invisible hemisphere), suggests that we are dealing with a more-or-less global instability pattern. In case of non-radial pulsations (Lobel et al. 1994; Fadeyev 2011), each hemisphere of the star should be well covered with a number of such oscillations moving up and down more or less in concert. The bumps and dips mentioned above could well be caused by this process.\n\n\\subsubsection{Semi-regular, weakly chaotic, or chaotic? }\n\nThe observations represented in Figs.\\,2 and 3 (representative for the other two objects as well) show that YHGs are subject to coherent sequences of pulsations. One can easily identify minima and maxima, although showing a variable shape from cycle to cycle, without significant secondary dips and peaks. Periods and amplitudes of individual cycles vary by no more than a factor two or three (eruptive episodes excluded). Semi-regularity (semi\\,=\\,half) is an often-used but vague description for the variability of the different types of evolved stars. Below we summarise two of the most interesting studies dedicated to modeling the pulsation sequences of late-type stars and compare these latter with those of YHGs.\n\nFollowing suggestions by Ashbrook et al. (1954) and Deeming (1970), Wisse (1979) used a continuous second-order auto-regression process to derive model light curves of red giants. The latter author distinguished two groups, one corresponding to strongly damped systems generating irregular (chaotic) model curves, and the other one consisting of slightly damped and undamped systems, generating light curves showing striking similarities with the real ones of Lb, SRb, and SRa giants; for example (i) the uniqueness of each cycle, (ii) a trace of a quasi-periodicity, and (iii) the presence of humps and shoulders. Wisse tentatively concluded that red giants are likely subject to random processes being dominant over one regular process.\n\nDifferently from their many predecessors, Icke et al. (1992) approached the study of such quasi-periodic behaviour with numerical studies of the dynamics of a driven one-zone stellar oscillation. These latter authors concentrated on the increase of the irregularity of the outer layers (the `mantle') of evolved AGB stars, which are low-mass stars. The more AGB stars evolve, the more mass they lose, and the less massive the mantle becomes, the more the oscillations become irregular.\n\nWe assume that the results of Wisse (1979) and Icke et al. (1992) are in general also valid for many other `semi-regular' convective variables such as YHGs, although these other variables are likely to be much more massive. Yellow Hypergiants have lost at least half of their initial mass, and are still highly eruptive on a timescale of decades, or less. Additionally, we assume that the mantle of YHGs will also have a lower mass compared to the total mass, similar to AGBs.\n\nThe calculations of the mantle motions by Icke et al. (which are assumed to be representative of the light curves), yield a number of global properties. A fine-looking series of oscillations may eventually, sometimes quite suddenly, switch to a different pattern.\nWe noted such sudden switches also in the mean quasi-periods of YHGs: after a number of oscillations, for example about half a dozen, the mean jumps to a smaller (or larger) one.\n\nWith respect to this type of intermittent behaviour, Icke et al.: state that if a stellar oscillator is found to be intermittent, an impractical amount of observing time is needed to establish the periodicity. This is exactly the case for YHGs (apart from the sudden switches of the mean quasi-periods mentioned above); each pulsation has a different duration. Additionally, we observe that also with respect to morphology, each pulsation is different as well. In short, each pulsation is unique, and predicting any property of the next pulsation is impossible.\n\nIcke et al. also concluded that stars with a large interior radius relative to that of the mantle likely show a more chaotic behaviour. For YHGs the opposite is likely the case: the outer layers are extremely extended, thus reducing the chaotic character.\n\nNoteworthy is that for the calculation of the model light curves of different kinds of evolved late-type stars, Wisse (1979) and Icke et al. (1992) did not take into account any potential long-term variation; Sect.\\,4.\n\nIn conclusion, we characterise the instability of the four objects studied here as weakly chaotic, that is, not as chaotic as the individual oscillations that are clearly separated from each other. In other words, the cadence of oscillations appears to be coherent.\n\n\n\\subsection{HR\\,5171A: a contact binary?}\n\nOne of the important purposes of Figs.\\,2 and 3 was to emphasise that despite the weakly chaotic character, the quasi-periodic sequences of YHGs show a coherent behaviour. Therefore, it was easy to conclude that HR\\,5171A does not show any trace of a second variable source as proposed by Chesneau et al. (2014). The light curve of these latter authors consists of a double-peaked brightness oscillation with two unequal amplitudes, and with a supposed binary period of about 1300 d. This is unlikely, as if this were the case Fig.\\,3 would not show any coherent succession of light oscillations, but a quite chaotic distribution of the observations. The resulting brightness fluctuations would vary from very small to twice the present size (Fig.\\,3), by two oscillations with slightly different amplitudes, but significantly different periods (the one with half the binary period: about 660 d, and the other one by pulsation, at the time about 500 d). With respect to the morphology of the observed light curves observed in Figs.\\,2 and 3, it happens to be that pulsations of $\\rho$\\,Cas and HR\\,5171A showed minima with alternating depths for some time. The selection of cycles as done by Chesneau et al. and depicted in their Fig.\\,8 was obviously sensitive to this bias, resulting in the double wave with two unequal minima. However, there are mitigating circumstances: at the time very little was known about this treacherous photometric property (hopefully, the current paper fills this gap).\n\nAnother argument in favour of pulsations only is that the monitoring of the V$_{\\rm rad}$ curve during two pulsations (no.\\,16 and no. 17 in Fig.\\,1 of Paper \\.I) by Lobel et al. (2015) showed a regular phase lag of about 0.4 with respect to the light curve. This is also typical for $\\rho$\\,Cas (Lobel et al. 2003).\n\nHowever, we do not deny the possible presence of some companion, albeit it perhaps only in the field of view of the Hipparcos satellite; see the photometric analysis by Eyer (1998) and footnote no.\\,2 in van Genderen et al. (2015). Based on AMBER\/VLTI observations, Chesneau et al. (2014) discovered a bright spot in front of the primary disk. The PIONIER\/VLTI interferometric observations by Wittkovski et al. (2017b, c) obtained impressive near-infrared (NIR) $H$-band images. They claim that the contact companion, suggested by Chesneau et al. (2014) is tentatively supported by a few images of HR\\,5171A showing no precise circular circumference. One image even showed a clear bubble in the SW direction, which according to these authors might be the companion. A tentative model for the orbit is proposed. Secondary features like bright patches are assigned to a convection cell.\n\nIt can also be concluded that if a low-mass contact companion to HR\\,5171A were present, any geometrical light variation due to deformation is obviously too small to be clearly detected.\nThe diameters of those few images mentioned above of HR\\,5171A look different from each other, obviously due to the pulsations (light curve at the time shown in Fig.\\,15), and the circumference is far from perfectly circular. This is understandable considering the very low surface gravity.\n\nTherefore, we also conclude that the well-covered and clearly plotted pulsation sequences like those in Figs.\\,2 and 3, are essential to understand the characteristic instabilities of YHGs and possibly to correct questionable interpretations.\n\n\\subsection{Inconsistencies between brightness and colour, and between spectroscopic and photometric temperatures.}\nThe analyses presented in the following sections have not been performed previously.\nWe embarked on the (tedious) traditional method of plotting many tens of individual pulsations per object on graphic paper, comprising thousands of data points (for light and colour curves). The intention was to get an idea about the degree of their variable morphological properties, whether quasi-periods and light amplitudes are different during hot and cool stages, and more importantly to try to understand why spectroscopic and photometric temperatures, T(Sp) and T(Phot), always differ. Pulsations during which a T(Sp) was obtained were particularly important for our purposes.\n\nWalker (1983), studying photometry and spectroscopy of HR\\,8752, was probably the first to notice that a relationship between the photometric magnitudes (and thus the continuum radiation) and the spectrum of YHGs was not always straightforward. He was not the only one.\nFor example, Arkhipova et al. (2009) noted ambiguous relations between brightness and colours of the HD\\,179821 pulsations during 2000--2009, and attributed them to increasing optical depth during high mass-loss episodes.\n\nWe also noticed that differences exist between the temperatures based on spectroscopy and photometry: large and small for high and low stellar temperatures, respectively. Such differences not only exist for $\\rho$\\,Cas, but also for the three other objects.\n\nThe purpose of the following sections is to enhance our understanding of the photometric properties and the probable cause of the fact that T(Sp) is usually higher than T(Phot).\n\n\\subsection{Correlation diagram $V$\/$(B-V)$: similarities and differences between hot and cool YHGs.}\nStudying the morphology of the series of detailed light and colour curves provides valuable information about the wavelength dependency of amplitudes, the variation of the duration of pulsations, the size of amplitudes, and the mean rate of variation of temperature and radius. Enhanced mass-loss episodes could well result in inconsistent behaviour between brightness and colour indexes, and consequently in unreliable stellar properties. Therefore, we were interested in the properties of the correlation diagrams $V$\/$(B-V)$.\n\nContrary to YHGs (including many RSGs), Pop. I Cepheids and RR\\,Lyrae stars are radially pulsating stars, and obey a number of fixed properties. Most of their light curves have steep ascending branches with respect to the descending ones. Maximum acceleration of the expansion velocity happens during this phase, and is responsible for a fast temperature rise (e.g. Pel 1978, Lub 1977, 1979). Therefore, at the same visual brightness $V$, $(B-V)$ is slightly bluer (T$_{\\rm eff}$ is higher).\n\nOur analysis of the four objects reveals a greater amount of photometric diversity than what is seen for the Cepheids and RR Lyrae stars with respect to the correlation diagrams. We noted an eye-catching difference between the $V$\/$(B-V)$ diagrams of the pulsations of the hot YHG HD\\,179821 (T$_{\\rm eff}$\\,=\\,6800\\,K, Arkhipova et al. 2009) and those of the cooler HR\\,5171A, $\\rho$\\,Cas, and HR\\,8752 when the latter was also as cool as its two sisters: about 5000\\,K (Nieuwenhuijzen et al. 2012).\n\nIntrinsic differences between the properties of light and colour curves between hot and cool stages of YHGs can be expected, which must be related to their evolutionary stage. Figure 1 shows that in the hot stage, HD\\,179821 and HR\\,8752 were crossing the Yellow Void (and HR\\,8752 even evolved into the Yellow-Blue Void) along a BL evolution: the stars contracted and the mean density increased. On the contrary, the YHGs  HR\\,8752, $\\rho$\\,Cas, and HR\\,5171A in the cool stage are moving along a RL just outside the Yellow Void (Fig.\\,1).\nIndeed, we discovered that light curves of YHGs HD\\,179821 and HR\\,8752  in the hot stage after the 1973 eruption show smaller visual amplitudes (0\\fm1--0\\fm2) and much shorter quasi-periods (100$^{\\rm d}$--200$^{\\rm d}$) than the cooler YHGs. Furthermore, about  30\\% of the pulsations, best documented for HD\\,179821, even showed shifted colour extremes with respect to the light extremes (thus, light and colour curves do not always run in phase). Sometimes the reddest colour is reached during the ascending branch, and the bluest colour during the descent, and vice versa.\n\nA very global conclusion is that we discovered that variable light curve properties like quasi-periods and amplitudes depend on whether the star is hot (periods are short, amplitudes are small), or cool (periods are longer, amplitudes are larger). We return to this subject in Sect.\\,3. To look for the cause of the temperature inconsistency, we have to use another amplitude ratio; see the following section.\n\n\\subsection{Ratios Ampl\\,$B$\/Ampl\\,$V$ and Ampl\\,$V$\/Ampl\\,$(B-V)$: influence of enhanced mass loss episodes.}\n\nOther possibilities to characterise the variability of pulsating stars are the amplitude ratios Ampl\\,$B$\/Ampl\\,$V$ (amplitudes in $B$ are always larger than in $V$) and Ampl\\,$V$\/Ampl\\,$(B-V)$ (for ascending and descending branches, which are mainly used here). Both ratios are a measure of the temperature change during the pulsation. The size of the amplitudes is independent of extinction and reddening corrections, and therefore so are their ratios.\n\n--{\\bf $\\rho$\\,Cas}: It appeared that for $\\rho$\\,Cas the mean ratio Ampl\\,$B$\/Ampl\\,$V$ for tens of ascending and descending branches of pulsations (or only parts of them if there is a lack of data points, and eruptions excluded) and based on the robotic data sets 2003--2017 (by G.W.H.) is 1.6\\,$\\pm$\\,0.3 (st.dev.). The corresponding ratio Ampl\\,$V$\/Ampl\\,$(B-V)$\\,=\\,1.7\\,$\\pm$\\,0.5 (st.dev.) It should be noted that in the 1986--2000 time interval, only $VRI$ photometry was performed. $(V-R)_{0}$ can easily be transformed into $(B-V)_{0}$ by dividing it by 0.75; see (eq.\\,1) of Sheffer \\& Lambert (1992).\nWe call these `normal' pulsations, as they are the majority. This ratio also applies to well-defined light curves of HR\\,5171A and HR\\,8752. We note that our designation `normal' is relative, and it only serves as a reference point.\n\n`Abnormal' pulsations also exist, with amplitude ratios significantly larger or smaller (rare) than the value of 1.7\\,$\\pm$\\,0.5 above. Between the two massive eruptions of $\\rho$\\,Cas in 1986 and 2000, many pulsations with overly large amplitude ratios occurred. That could be established because the star was monitored photometrically (1986--2000, in $UBV$ and $VRI$) and spectroscopically (1993--2002, by Lobel et al. 2003). Figure\\,1 of Lobel et al. 2003 shows the $V$ and visual light curves and the V$_{\\rm rad}$ curves, while the dates of the spectra and the presence of emission lines are indicated.\nReferences used for this analysis are listed in Appendix D: 2.6.\nThe fact that the ratios Ampl\\,$V$\/Ampl\\,$(B-V)$ are higher than 2.2 up to about 5, and also the relatively large variation of these ratios within this group, is a warning that some irregular phenomenon has disturbed the intrinsic photometric parameters $B$ and $V$. The height of the ratios means that the denominator is significantly smaller than for the `normal' ones, pointing to some selective absorption. Therefore, $(B-V)$ curves show overly small amplitudes, and the $(B-V)$ values are too red, causing the overly small temperature ranges.\n\nTwo such pulsations, i.e., those in 1993 ($\\sim$\\,JD\\,24\\,49300) and 1998\\, ($\\sim$\\,JD\\,24\\,51065), with relatively high spectroscopic temperatures (T(Sp)\\,=\\,7250\\,K), high maxima, and showing emission lines in the spectrum due to enhanced mass loss episodes, had ratios of 5.1 and 2.5, and 4.4 and 4.1, for ascending and descending branches, respectively (spectroscopy by Israelian et al. 1999; Lobel et al. 2003). The pulsation in 1997 ($\\sim$\\,JD\\,24\\,50575) however, that is in between the two above, without emission lines, and a with maximum rivaling the couple above, had abnormal ratios as well, 4.4 and 5.8, indicating that the absorption was still present.\n\nLastly, the ratio for the ascent to the minimum ($\\sim$\\,JD\\,24\\,51320) preceding the 2000-eruption max.\\,1 ($\\sim$\\,JD\\,24\\,51590) is also too high: 2.5, while no obvious emission was reported. The spectroscopic temperature rose to 7600\\,K, and then the brightness declined to a deep minimum, indicating that $\\rho$\\,Cas expanded and cooled to about $\\sim$\\,4400\\,K.\n\nDe Jager (1998, Ch.8) suspected that pulsations during an enhanced mass-loss episode show larger light amplitudes. This appeared to be indeed the case for all eruptive maxima. However, ordinary pulsations sometimes show relatively large amplitudes during line-emission periods (e.g. 1993 and 1998), but not always like those observed in 1997 and 1998.\n\n--{\\bf HD\\,179821}: Amplitude ratios were derived for most of the light curves of HD\\,179821. A few detailed pulsations are shown for example in Fig.\\,2 of Arkhipova et al. (2009) and in Figs.\\,1, 3, and 4 of Le Coroller et al. (2003). Their light curves also include observations by ASAS, Hrivnak (2001), and Hipparcos (ESA, 1997).\n\nJust like in the case of $\\rho$\\,Cas, we also identified `normal' and `abnormal' pulsations for HD\\,179821 in the two time intervals 1991--1999 (JD\\,24\\,48500--JD\\,24\\,51500) and 2000--2006 (JD\\,24\\,51700--JD\\,24\\,53700) on account of the ratios. Smaller ratios than 2.2 belong to the `normal' ones, as they are the majority as well.\nThe mean Ampl\\,$B$\/Ampl\\,$V$\\,for HD\\,179821 is\\, 1.6\\,$\\pm$\\,0.3 (st.dev., n\\,=\\,14), thus similar to that of $\\rho$\\,Cas (1.6\\,$\\pm$\\,0.3). One peculiar pulsation, was excluded: the one of $\\sim$\\,JD\\,24\\,52500. Consequently, the mean ratio Ampl\\,$V$\/Ampl\\,$(B-V)$\\,=\\,1.7\\,$\\pm$\\,0.5 for HD\\,179821 is similar to the one for $\\rho$\\,Cas.\n\nAt least six pulsations of HD\\,179821 are abnormal (the cases with light and colour curves running out of phase were excluded, as mentioned in Sect.\\,2.5., although they are abnormal as well). Indeed, often the H$\\alpha$ profiles of HD\\,179821 offered proof for enhanced mass-loss episodes (Tamura \\& Takeuti 1991; Zacs et al. 1996); see also Arkhipova et al. (2001).\nWe would like to mention one concrete case with Ampl\\,$V$\/Ampl\\,$(B-V)$\\,=\\,6.2, and coinciding with an enhanced mass-loss episode based on the H$\\alpha$ profile, observed at JD\\,24\\,52793, by Sanchez Contreras et al. (2008).\n\n--{\\bf HR\\,8752}: This object showed a connection between enhanced mass-loss episodes and abnormal amplitude ratios during its cool YHG stage, especially between 1976 and 1981. At the time, Smolinski et al. (1989) identified emission features in the spectrum (see Figs. 9 and 10 of de Jager 1998), while $BV$ photometry indicated overly small $(B-V)$ amplitudes (caution is called for: the number of data points was at times rather low). The photometric observations were conducted by Moffett \\& Barnes 1979; Percy \\& Welch 1981; Walker 1983: Arellano Ferro 1985; Zsoldos \\& Olah 1985).\n\nOnly for one light curve of HR\\,8752 in 1976 (thus, after the 1973 eruption) were the maximum ($\\sim$\\,JD\\,24\\,43100) and the following minimum ($\\sim$\\,JD\\,24\\,43400) properly defined by $B$ and $V$ data points, for which the amplitude ratio was relatively abnormal pointing indeed to an enhanced mass-loss episode (Smolinski et al. 1989): Ampl\\,$V$\/Ampl\\,$(B-V)$\\,=\\,6.7.\n\nIn conclusion, we discovered that for YHGs, overly high Ampl\\,$V$\/Ampl\\,$(B-V)$ values signal instable atmospheric conditions, sometimes supported by the presence of emission lines in the spectrum. The overly low $(B-V)$ amplitudes are likely caused by some gas layer with an enhanced opacity, and showing variability during the oscillations. The absorption is obviously wavelength dependent: more for $B$ (a rough estimation: of the order of 0\\fm1 up to one magnitude. The higher the temperature, the higher the absorption) than for $V$, but the extinction law remains unknown. Furthermore, such a layer seems to be able to survive some successive pulsations, considering the continuation of abnormal amplitude ratios for pulsations with no noticeable emission lines.\n This appears to be the case for the time interval 1986-2000 of $\\rho$\\,Cas and for HD\\,179821, considering the occurrence of a number of pulsations with abnormal amplitude ratios, while no emission lines were present in the spectra.\n\nA sketch of a few fictitious pulsations in $B$ and $V$ is presented in Fig.\\,D.1. of Appendix\\,D:2.6.: to illustrate this cyclic behaviour with the aid of three sets of amplitudes and their ratios. More evidence for a difference between T(Sp) and T(Phot) is given in Appendix\\,E:2.6.\n\n\n\\subsection{The temperature scales: calculations of stellar properties.}\n\nIf one would like to derive stellar properties from the photometry of many types of stars, such as the temperature (e.g. to compute M$_{\\rm bol}$, the radius R and its variation), a short discussion is needed on a number of temperature calibrations for all stars based on broadband continuum photometry.\n\nThe Schmidt-Kaler (SK) calibration (1982) uses the $(B-V)_{0}$ to derive the T$_{\\rm eff}$ and the BC and is valid for stars up to supergiants of luminosity class Iab only, simply because at the time there was a lack of well-calibrated super- and hypergiants.\nIf one uses the SK calibration of Iab stars to derive temperatures for hypergiants, they will be too high. Therefore one should use the de Jager-Nieuwenhuijzen (dJN) calibration (de Jager \\& Nieuwenhuijzen 1987); see Appendix\\,F:2.7.\n\n\\subsection{$\\rho$\\,Cas pulsations and the definition for long-term variations.}\n\nDespite the fact that stellar parameters from YHGs derived from photometry are expected to be unreliable (Sects.\\,2.4--2.6), to start with the temperature derived from $(B-V)$, it would be of most importance to start a photometric and spectroscopic monitoring program (daily) for individual pulsations.\nIn this way the trend of both temperatures can be studied, the purpose being to learn more about the absorption law. It would be even more rewarding if the mass loss rate were monitored as well, so that the explanation of the `cyclic absorption variation' (Appendix, Fig\\,D.1.) can be verified.\n\nAs  only  high-quality photometry is required for such a program, like the multi-colour photometry of Henry (1995, 1999; see Sect.\\,2.2), we selected two fine individual pulsations from the $\\rho$\\,Cas sequence in Fig.\\,2, that is, Figs.\\,4 and 5;\nsee Appendix\\,G: 2.8. for an explanation of why spectroscopic monitoring is also indispensable.\n\nFigure\\,2 shows the 2003--2017 $BV$ sequence of quasi-periods (bright is up). The magnitudes $V$ and $B$ are relative to the comparison star HD\\,223173 ($V$\\,=\\,5.513, $B$\\,=\\,7.160). Similar high-quality observations made recently by one of the current authors (EJvB) are represented by dots on the very right (see Appendix B: 2.1).\n\nLong-term variations are the mean curves sketched through the median brightness and colour indices of all pulsations. Although already known and discussed by many researchers for decades, we are the first to suggest this definition.\nThe numbers placed above individual $B$ pulsations in Fig.\\,2 refer to pulsations which are mentioned in this paper.\nThe LTV in $V$ runs almost constant over time, contrary to the one in $B$; it shows a wavy trend, with an amplitude of $\\sim$\\,0\\fm2. This means that the individual pulsations in the maxima of the LTV ($B$) are on average bluer and hotter than in the minima.\n\nAdditionally, the LTV in $B$ also shows a significant dip (including pulsation no.\\,2, depicted in Fig.\\,4), lasting $\\sim$\\,300$^{\\rm }$ d, but in $V$ it is hardly visible. A similar type of relatively fast variation, independent of the pulsations, was identified in the light variations of the other three YHGs.\n\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{img020.pdf}}\n \\caption{ Light curve no.\\,2 from Fig.\\,2 of $\\rho$\\,Cas relative to the comparison star (in magnitudes; bright is up).  On top the light curve is remarkably dome-shaped. Dots are individual data points; the scatter is extremely small: 0\\fm01--0\\fm02. The dashed curve, part of the ascending branch, is regrettably not observed. The relatively large scatter of a number of data points preceding the dome near JD\\,24\\,55480 is due to unstable sky conditions as the reference stars showed this scatter as well. Two spectral temperatures by Klochkova et al. (2014) are indicated.}\n  \\label{Fig4}\n \\end{figure}\n\nFigs.\\,4 and 5 show the finely detailed $V$ and $B$ light curves relative to the comparison star, with the extremely small scatter of 0\\fm01--0\\fm02. One can make out the slow contraction phase to maximum light and then the expansion phase (only partly) to minimum light of a very low-gravity photosphere with a huge dimension of a few astronomical units.\n\nWe selected Fig.\\,4 because of two spectral temperature T(Sp) determinations, made by Klochkova et al. (2014) (JD\\,24\\,55409 and JD\\,24\\,55463). One coincides with a gap in the time-series (dashed curves) representing the minimum, and therefore interpolated magnitudes are used below, and the second one coincides with the plateau. Below, T(Sp) is compared with the calculated photometric temperatures based on the SK and dJN methods using the observed $(B-V)$ obtained at the same time.\n\nThe ascending branch in Fig.\\,4 shows a plateau and a peculiar symmetrically dome-shaped maximum lasting 37$^{\\rm }$ d with amplitudes of 0\\fm08 and 0\\fm05 in $B$ and $V$, respectively. Subsequently, the branch shows a decline almost equal to the rise, and then a plateau. After a gap in the time-series, a new pulsation started, but brighter than the previous one according to Fig.\\,2.\n\nThe pulsation in Fig.\\,5 (with no T(Sp) determinations), labelled no.\\,3 in Fig.\\,2, showed almost the same surprising light-curve morphology. The duration of the dome-shaped maximum lasts longer, namely 65$^{\\rm }$ d, and the amplitudes are larger: 0\\fm115 and 0\\fm070 in $B$ and $V$, respectively. The ascending branch is slightly bumpy (which is intrinsic) with a timescale of weeks.\n\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{2019_RHOCASfin2_BV2.pdf}}\n   \\caption{Similar to Fig.\\,4: but for light curve no.\\,3 of Fig.\\,2 of $\\rho$\\,Cas relative to the comparison star, also dome-shaped like in Fig.\\,4. Dots are individual data points. The ascending branch shows a number of significant bumps with a timescale of a few weeks.}\n   \\label{Fig.5}\n \\end{figure}\n\n\n A great variety of secondary features exist in other pulsations of $\\rho$\\,Cas, that is, no.\\,5 (Fig.\\,2), which has a plateau in the ascending branch lasting almost two months, and no.\\,6 (Fig.\\,2) shows a maximum that is regrettably interrupted by a gap in the time-series and is flat and lasts no less than four months. One can only speculate as to the causes of all these features.\n\nTable\\,2 summarises the calculated\nstellar properties for the selected locations in Fig.\\,4. As we assume that the plateaus represent constant temperature, radius, and so on, we have chosen the start and the end of the plateau, together with the minimum and the maximum.\nThe physical properties are horizontally sorted. Below the double line, we show the differences in time, $\\triangle$\\,JD, and the $V_{0}$ and $(B-V)_{0}$, the T(Sp) and the two calculated temperatures for Iab stars based on the SK calibration and for Ia$^{\\rm +}$ stars using the dJN calibration.\n\nTable\\,3 summarises for Fig.\\,4, in different temperature columns, the BC, M$_{\\rm bol}$, and the radius R (in R$_{\\rm \\odot}$) derived from the formula:\nlog\\,R\/R$_{\\rm \\odot}$\\,=\\,8.47 - 0.2M$_{\\rm bol}$ - 2log\\,T$_{\\rm eff}$. The symbol\n$\\triangle$\\,R represents the decrease, or increase of R (- or +, respectively) with respect to the previous location, and the V$_{\\rm rad, puls}$ is in kms$^{-1}$ (+ for contraction and - for expansion) between two successive locations (vertically sorted). We note that they represent the mean velocity between the two locations. We used the accepted distance of 3.1\\,kpc.\n As YHGs are in general non-radial pulsating stars (Sect.\\,2.2.), the $\\triangle$\\,R, and V$_{\\rm rad,puls}$ are lower limits.\n\n\\begin{table}\n\\caption{Extinction free visual brightness, colour indices and three types of temperature determinations for four instances in time in the ascending branch (two for the plateau: the first for the beginning and the second for the end of the plateau) of a pulsation of $\\rho$\\,Cas shown in Fig.\\,4. We note that T(SK) and T(dJN) are based on the $(B-V)_{0}$, therefore, they are photometric temperatures T(Phot).}\n\\label{table:2}\n\\centering\n\\begin {tabular}{c|c|c|c|c}\n\\hline\\hline\nLocation & min & plateau & plateau & max \\\\\n\\hline\\hline\nJD-- & 5409 & 5464 & 5496 & 5515 \\\\\n245\\,0000 & & & & \\\\\n\\hline\\hline\n$\\triangle$\\,JD & -- & 55\\,d & -- & 19\\,d \\\\\n$V_{0}$ & 3.28 & 3.12 & 3.13 & 3.07  \\\\\n$(B-V)_{0}$ & 0.79 & 0.73 & 0.72 & 0.70 \\\\\nT(Sp) & 5777\\,K & 6044\\,K      \\\\\nT(SK)  & 5450\\,K & 5640\\,K & 5660\\,K & 5710\\,K  \\\\\nT(dJN) & 4890\\,K & 4980\\,K & 5000\\,K & 5040\\,K \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n \\caption{BC, M$_{\\rm bol}$, and radius R for four locations in the ascending branch (two for the plateau: the first for the beginning and the second for the end) of the pulsation of $\\rho$\\,Cas shown in Fig.\\,4, the radius variation $\\triangle$\\,R and contraction\/expansion velocity $V_{\\rm rad, puls}$. The latter represents the mean velocity between the selected locations. We note that they represent pure radial velocities of the continuum radiating layer, and are independent of the space velocity. }\n \\label{table:3}\n \\centering\n \\begin{tabular}{c|c|c|c}\n  \\hline\\hline\n  & T(Sp) & T(SK) & T(dJN) \\\\\n  & {\\bf min} & {\\bf min} & {\\bf min} \\\\\n \\hline\\hline\n & 5777\\,K & 5450\\,K & 4890\\,K  \\\\\n \\hline\n  BC & -0.13 & -0.16 & -0.20 \\\\\n  M$_{\\rm bol}$ & -9.31 & -9.35 & -9.39 \\\\\n  R &     668\\,R$_{\\rm \\odot}$ & 779\\,R$_{\\rm \\odot}$  & 970\\,R$_{\\rm \\odot}$ \\\\\n \\hline\\hline\n  & T(Sp) & T(SK) & T(dJN) \\\\\n  & {\\bf plateau} & {\\bf plateau} & {\\bf plateau} \\\\\n \\hline\\hline\n & 6044\\,K & 5640\\,K & 4980\\,K \\\\\n \\hline\n BC & -0.10 & -0.14 & -0.19 \\\\\n M$_{\\rm bol}$ & -9.43 & -9.48 & -9.53 \\\\\n R & 636\\,R$_{\\rm \\odot}$ & 747\\,R$_{\\rm \\odot}$ & 981\\,R$_{\\rm \\odot}$ \\\\\n$\\triangle$\\,R & -22\\,R$_{\\rm \\odot}$ & -7\\,R$_{\\rm \\odot}$ & +27\\,R$_{\\rm \\odot}$ \\\\\n V$_{\\rm rad, puls}$ & +3.2\\,kms$^{-1}$ & +1.0\\,kms$^{-1}$ & -4.0\\,kms$^{-1}$ \\\\\n \\hline\\hline\n & T(Sp) & T(SK) & T(dJN) \\\\\n & {\\bf plateau} & {\\bf plateau} & {\\bf plateau} \\\\\n \\hline\\hline\n  &  & 5660\\,K & 5000\\,K \\\\\n  BC & & -014 & -0.18  \\\\\n M$_{\\rm bol} $ & & -9.46 & -9.51  \\\\\n R &  & 735\\,R$_{\\rm \\odot}$ & 964\\,R$_{\\rm \\odot}$ \\\\\n \\hline\\hline\n  & & T(SK) & T(dJN) \\\\\n  &  & {\\bf max} & {\\bf max} \\\\\n \\hline\\hline\n  & & 5710\\,K & 5040\\,K \\\\\n \\hline\n BC & & -0.13 & -0.17  \\\\\n M$_{\\rm bol}$ &  & -9.52 & -9.56  \\\\\nR & & 742\\,R$_{\\rm \\odot}$ & 971\\,R$_{\\rm \\odot}$ \\\\\n$\\triangle$\\,R &  & +7\\,R$_{\\rm \\odot}$ & +7\\,R$_{\\rm \\odot}$  \\\\\nV$_{\\rm rad, puls}$ &  & -3.0\\,kms$^{-1}$ & -3.0\\,kms$^{-1}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n The summary below lists the average differences between the calculated temperatures and radii based on different temperature scales.\nThe average differences in T and R according to Tables\\,2 and 3 and based on two temperature scales are:\nT(Sp)--T(dJN)\\,$\\sim$\\,990\\,K,  R(Sp)--R(dJN)\\,$\\sim$\\,-300\\,R$_{\\rm \\odot}$.\n\nAccording to Klochkova (2014) and Klochkova et al. (2014), the uncertainties in T(Sp), based on ratios of selected spectral lines being sensitive temperature indicators, are in the range of 40\\,--160\\,K only. Considering the temperature differences we report, as shown in Tables\\,2 and 3, these differences are negligible. Therefore, the differences between T(Sp) and T(Phot) (=\\,T(dJN) and T(SK)) in Tables\\,2 and 3 are due to the effect we discovered, and for which we offer an explanation in Appendix\\,Fig.\\,D.1.\n\nObviously, the observed $(B-V)$ is too red, and is therefore inappropriate for deriving accurate temperatures and all other stellar properties.\nFor example, the trend of the calculated photometric radii at the four locations in Fig.\\,4, especially for the YHG calibration, are incorrect. They clearly run opposite to radius variations of YHGs, and should decrease towards maximum brightness (=\\,contraction, plus sign) instead of increasing. Its ratio Ampl\\,$V$\/Ampl\\,$(B-V)$\\,=\\,2.3, meaning that the $B$ amplitude with respect to $V$ amplitude is indeed too small (Sect.\\,2.6.). This suggests that absorption increases with temperature: at maximum light the absorption of $B$ increased with respect to the absorption in $V$.\n\nThe V$_{\\rm rad}$ value of 3.2\\,kms$^{-1}$ (second column), which is based on spectroscopy and should be multiplied by 1.4 to correct for projection effects, and so on, could become of the right order if the overly low radius variation (for which the correction is unknown) is also taken into account: Lobel et al. 2003: 5--10\\,kms$^{-1}$ and Klochkova et al. 2014: 7\\,kms$^{-1}$.\n\nThe above is in support of the existence of a cyclic absorption behaviour (Appendix\\,Fig.D.1.): as a consequence, the photometry of $\\rho$\\,Cas (and of all YHGs) does not represent a normal YHG photosphere. Stellar properties based on photometric data weakened by absorption with an unknown absorption law are unreliable. Again, we estimated that the absorption in $B$ is of the order of 0\\fm1 up to a few times this value; in $V$ the absorption is presumably about a factor two smaller, but that is very uncertain. The higher the mass-loss rate and the higher the temperature, the higher the absorption.\n\nThe photometric parameters of the pulsation in Fig.\\,5 (no spectral temperatures known), of which the amplitude ratio is 1.2, and thus `normal', are less inconsistent, yet the calculated temperatures (dJN) are still too low by hundreds of degrees with respect to the T(Sp) in Fig.\\.4, while both pulsations do not differ much with respect to the medium brightness. Although the photometry of the other YHGs are less accurate, our conclusions were similar. Therefore, we suspect that a high-opacity layer is always present in the atmospheres of YHGs, and its absorption capacity depends on the density of that layer (higher by enhanced mass loss) and on the temperature (higher by a temperature rise), and vice versa.\n\nThe magnitudes of $\\rho$\\,Cas, used in Tables\\,2 and 3 offer a chance to calculate errors due to scatter as small as $\\pm$\\,0\\fm01. The result is approximately $\\pm$\\,15\\,K and $\\pm$\\,10\\,R$_{\\rm \\odot}$.\nThe relatively large size of these errors demonstrates the extreme sensitivity  of the calculated stellar properties to `very small' photometric errors.\nOur message is that this analysis underlines the importance of a long-lasting photometric and spectroscopic monitoring all over the world; see Appendix\\,G: 2.8.\n\n\\subsection{Proposed distance reductions for $\\rho$\\,Cas and HR\\,5171A.}\nBelow we propose shorter (conventional) distances for $\\rho$\\,Cas and HR\\,5171A. We ignore the distances based on parallaxes by the satellites Hipparcos and Gaia (Sect.\\,1).\n\nThe size of the radii derived from the spectroscopic and radial velocity\nmonitoring campaign during the eruption in the year 2000  motivated us to question whether the accepted distance of $\\rho$ Cas by Zsoldos \\& Percy (1991) of 3.1\\,kpc\\,$\\pm$\\,0.5\\,kpc is too large. We note that the sizes of the radii obtained in this way were derived independently of the distance by Lobel et al. (2003). They estimated that the extreme dimensions of the stellar radius at the eruption maximum and at the deep minimum brightness were $\\sim$\\,400\\,R$_{\\rm \\odot}$ and $\\sim$\\,1000\\,R$_{\\rm \\odot}$, respectively.\n\nThe calculated mean radius of an average pulsation with a normal amplitude in $V$ of for example  0\\fm21, is $\\sim$\\,960\\,R$_{\\rm \\odot}$ (dJN), which is too close to the radius for the deep eruption (photometric) minimum (when the star has reached its maximum size) to be plausible. Radii based on the SK calibration are about 200\\,R$_{\\rm \\odot}$ smaller, but this is definitely only valid for SGs of type Iab. If de Jager \\& Nieuwenhuijzen (1987) were found to have assumed overly extreme properties for hypergiants relative to the Iab SGs, then a reduction of the radius difference above from 200\\,R$_{\\rm \\odot}$ to say 100\\,R$_{\\rm \\odot}$ would yield a radius of 860\\,R$_{\\rm \\odot}$ (instead of the above 960\\,$_{\\rm \\odot}$), which would still be too high to be credible. Therefore, the error on the dJN radius cannot be much more than that on the SK radius.\n\nOther options are to decrease the interstellar extinction from 1\\fm44 (Table\\,1) to 1\\fm0 for example, and the reddening from 0\\fm45 to about 0\\fm30, but these values are not likely. Our choice of 2.5\\,kpc\\,$\\pm$\\,0.3\\,kpc is rather arbitrary. The M$_{\\rm bol}$ becomes about 0\\fm45 less negative (luminosity lower), and the calculated radii are reduced by a factor of $\\sim$\\,1.2: $\\sim$\\,770\\,R$_{\\rm \\odot}$, as well as the radius variations and the radial pulsation velocities. For us, these values are more acceptable with respect to the derived extreme dimensions above.\nThis case serves as a reminder that our attempts to obtain more precise information on YHGs are seriously hampered as long as distances remain uncertain, not only for $\\rho$\\,Cas, but also for HR\\,5171A below, and HD\\,179821.\n\nThe accepted distance of HR\\,5171A is 3.6\\,kpc, assuming that it belongs to Gum48d and that HR\\,5171B, at a distance of 10\" is a nearby companion: optically or physically (Humphreys et al. 1971; Schuster 2007; Schuster et al. 2006). However, subsequently M$_{\\rm bol}$\\,=\\,-10 -- -11 depending on the reddening (1\\fm0--1\\fm4) and reddening law used (3.1--3.5), creates a problem, as this is much too bright for a hypergiant. As far back as 2013 one of us (HN) emphasised that the accepted distance is too large. As a result, the radius would then also be much too large: between 6 and 12\\,AU. We think that this is too big, until new outcomes are undisputable (van Genderen et al. 2015).\n\nA distance of between 1 and 2\\,kpc for example, would entail a luminosity that is more in accordance with its Ia$^{+}$ hypergiant character, for example with M$_{\\rm bol}$ lying between -8.7 and -9.6 (Sect.\\,1) and a radius between 3 and 5\\,AU. We do not think that HR\\,5171A is a RSG as advocated by Wittkovski et al. (2017a); see Sect.\\,2.2. HR\\,5171A shows too many similarities with $\\rho$\\,Cas for example, like its eruptive activity, and above all, the spectra of the Big Three are almost identical. However, the H$\\alpha$ line of HR\\,8752 strongly differs with respect to the shape because of the permanent presence of a strong emission line.\nThus, the physical properties of the winds of the Big Three are in many respects identical (Lobel et al. 2015); in any case, RSGs do not show eruptive events.\n\nA portion of the light curve between JD\\,24\\,47200 and JD\\,24\\,48800 (1988--1992) was spectroscopically monitored, resulting in a well-covered V$_{\\rm rad}$ curve with an amplitude of 10\\,kms$^{-1}$ (Lobel et al. 2015). This correlated very well with the $V$ magnitudes between the maxima of pulsations\\,16 and 17 (numbered according Fig.\\,1 in Paper\\,I), indicating a steady contraction towards maximum brightness and the reverse to the brightness minimum.\nWe note that at the time HR\\,5171A had reached its faintest brightness: $V$\\,$\\sim$\\,7\\fm0 and reddest colour so far: $(B-V)$\\,$\\sim$\\,2.6 (Figs.\\,13 and 14).\nAppendix\\,H.2.9. presents a description of a method which is in principle suitable for deriving the distance of a pulsating star with the aid of the calculated V$_{\\rm rad puls}$ and the spectroscopically obtained V$_{\\rm rad}$, but $B$ and $V$ magnitudes of the pulsation cycle should be undisturbed.\n\nAnother argument favouring a much shorter distance than 3.6\\,kpc is based on the fact that at this distance, the position of HR\\,5171A would be below the Humphreys-Davidson limit on the theoretical HR diagram (shown in Fig.\\,6 of Paper\\,I) instead of far above it. At below 3.6 kpc, the distance with respect to the semi-empirical P\\,=\\,constant line for $\\sim$\\,500 d (the mean quasi-period of HR\\,5171A between the 1960s and 1990s; van Leeuwen et al. 1998) would be more in accordance with the positions of the other variable supergiants.\n\nFurther, the variability of the pulsation quantity Q\\,=\\,P$\\rho^{-1\/2}$ appeared to be compatible with stability studies of stellar models and was first described by Maeder \\& Rufener (1972). Based on that study, Burki (1978) derived a formula between the dependency of the quasi-period P on the mass M\/M$_{\\rm \\odot}$, M$_{\\rm bol}$ and on T$_{\\rm eff}$.\nApplying in his Eq.(5) the approximate but plausible input parameters 25\\,M\/M$_{\\rm \\odot}$, -8.8, and 4300\\,K, respectively, the quasi-period becomes P = 473$^{\\rm d}$, that is, roughly of the same order as the observed mean quasi-periods (see Fig.\\,13). By varying the input parameters by trial and error, say by $\\pm$\\,5\\,M\/M$_{\\rm \\odot}$, $\\pm$\\,0\\fm2 and $\\pm$\\,200\\,K, respectively, in either directions, results are of the same order.\n\nA third argument for a shorter distance refers to the energy budget of the Gum48d nebula.\nSchuster (2007) considered HR\\,5171B as the single central engine of the associated HII region RCW80, just like Karr et al. (2009). Based on calculations on the present energy budget, the latter concluded that the nebula only needed one single ionizing O-type star, that is, the present B-type star HR\\,5171B, only a few million years ago still on the main sequence, just like HR\\,5171A at the same time. In other words, for the present energy budget of Gum48d, the presence of HR\\,5171A as a member of Gum48a is superfluous.\nIf HR\\,5171A is indeed a foreground star, a new place of birth should be found.\n\nOn the contrary, Humphreys et al. (1971) and Schuster (2007) offered a number of arguments based on various independent techniques and physical considerations favouring the larger distance. Therefore, the distance remains debatable for the time being.\n\nIn conclusion, if our preferred  distance of 1.5\\,kpc\\,$\\pm$\\,0.5\\,kpc were found to be correct, it would dethrone HR\\,5171A as one of the biggest stars known (Chesneau et al. 2014; Wittkovski et al. 2017c).\n\nThis distance reduction would also reduce the flux of the reported Blue Luminescence by PAH molecules in the 1970s (van Genderen et al. 2015) by a factor $\\sim$\\,6.\nThis is because the excesses were measured relative to the stellar flux in the $L$ channel and would stay unaffected for both scenarios (1) and (2): i.e. the source of the excess lies in Gum48d, or in the outer envelope of HR\\,5171A, respectively.\nHR\\,5171A possesses an optically thick extended molecular and dust envelope at about 1.5 stellar radii (Wittkovski et al. 2017a). According to studies by Gorlova et al. (2009) and Oudmaijer \\& de Wit (2013) a dense optically thick layer, formed by the wind just above the photosphere is able to shield the stellar radiation as well. Thus, there are ample possibilities to locate neutral PAH molecules to become excited to the upper electronic state by high energetic photons (3.5--5eV, A.N. Witt 2013, priv.comm). There after, these molecules recombine and emit within a near-UV band (coinciding with the $L$ band of the Walraven $VBLUW$ photometric system) radiation, called blue luminescence, discovered by Vijh et al. (2004) in the Red Rectangle nebula.\n\n\\section{The long-term variations}\n\n \\subsection{Light and colour curves, and trends of light amplitudes and quasi-periods.}\n In this section we outline the photometric properties of LTVs (on which the pulsations are superimposed and that was defined in Sect.\\,2.8.) exhibited by the four selected objects. Our interest is also focussed on their possible influence on pulsations and the occurrence of eruptions which has never been explored before.\n\nLight and colour curves of LTVs often appear whimsical with timescales of hundreds up to a few thousand days. The evolutionary models usually reveal straight tracks on the HR diagram. Yet, we wondered whether LTVs represent in fact the evolutionary tracks, which once transformed into luminosity and temperature might not\nbe straight tracks on the HR diagram.\n Whether straight or irregular, the subsequent zigzag movements of the YHGs on the HR diagram are supposed to be part of the need to loose enough mass for the final course to the blue (Sect.\\,1, Fig.\\,1).\n\n These LTVs have already been noted and discussed by numerous researchers and observers in the past for HR\\,5171A  (see for references Appendix\\,I:3.1.), but generally they were reticent about a possible evolutionary origin, contrary to our present suspicions based on the thorough study of HR\\,8752 from 1850 until 2005, including the eruptive episode from 1973 by Nieuwenhuijzen et al. (2012). These latter authors convincingly showed that the long-term variations of HR\\,8752 shown in our Figs.\\,6 and 7 are evolutionary tracks, or loops: a red one until the eruptive episode in 1973, and then a blue one (Figs.\\,5, 6, 10, etc. in Nieuwenhuijzen et al. 2012). These evolutionary loops (RL and BL) represent the red and blue tracks (no.\\,3 and no.\\,4) of the models of Meynet et al. (1994), respectively.\n\nThe most obvious conclusion from the study by Nieuwenhuijzen et al. is that the LTVs of the other three objects should have something to do with evolution as well. To prove that conjecture, we scrutinized the literature for scattered photometric observations of pulsations from which the LTV  at the time could be derived. More or less complete light and colour curves were put together (thus curves sketched through the median brightness and colour). As this has never been done before, and because of the importance of these curves we show them in their entirety in Figs.\\,6--14. In Tables\\,4 and 5 of Sect.\\,3.2 we tabulate the characteristic zigzag changes of brightness and colours, still referred to as `evolutionary modes', and their timescales.\n\nLarge empty intervals in time series, portions with bad sampling, and the presence of eruptive phases decrease the reliability of sketched LTVs, but generally they cannot be grossly in error. In the case of scattered historic visual or photographic observations of stars dating back to 1850, like in the case of HR\\,8752 (Luck 1975; Arellano Ferro 1985; Zsoldos 1986a, b; Nieuwenhuijzen et al. 2012), we connected individual data points and the averages of small clusters of data points.\n\nBecause of their importance these curves merit careful description.\n\n{\\bf --HR\\,8752:} Figures\\,6 and 7 depict the LTVs for HR\\,8752 in $V$, $B,$ and $U$, and $(B-V)$ and $(U-B)$, respectively, for the time interval 1941--1994.\n\nBased on a collection of numerous historical data since 1840, and modern photometric data of HR\\,8752, Nieuwenhuijzen et al. (2012) concluded that HR\\,8752 evolved through an increasing reddening and cool episode in the 1960s and 1970s. It is noteworthy that the corresponding red minimum in our Fig.\\,7 is also obvious in the log\\,T$_{\\rm eff}$\/JD diagram of Fig.\\,10 of Nieuwenhuijzen et al. (2012). These latter authors suggested that a massive eruption should have happened around 1973, although photometry was absent.\n\nTheir assumption was based on spectroscopic observations made at the time.\nLuck (1975) derived from spectral scans a temperature of 4000\\,K in August 1973, and $(B-V)$\\,=\\,1.78 in November 1974 but alas, by the scarcity of photometric $B$ and $V$ magnitudes, we could not calculate T(Phot), despite the expected amplitude of $\\sim$\\,1\\fm0. This can be explained by the fact that the gaps in the time-series are much larger than the duration of an eruption: 400$^{\\rm d}$--700$^{\\rm d}$.\n\nAdditionally, we found out that light amplitudes and quasi-periods of HR\\,8752 since 1976 decreased linearly until 1993 to about 0\\fm05--0\\fm1 and 100$^{\\rm d}$--150$^{\\rm d}$, respectively. This is a consequence of the continuous contraction of the star and the increase of its atmospheric density.\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{2019_FIG1fin_HR8752.pdf}}\n  \\caption{Long-term variables of HR\\,8752 in $V$, $B,$ and $U$ in  1942--1994. The arrows at the bottom indicate dates of spectroscopic observations by Smolinski et al.\n  (1989), taken in 1969 and 1970, and between 1979 and 1982; see also\n    de Jager (1998, Figs.\\,9 and 10 in Sect.\\,7). These spectroscopic observations signalled the presence of an enhanced mass-loss episode and a massive eruption in 1973 (arrow), which was not photometrically recorded\n  (Nieuwenhuijzen et al. 2012). We note that after 1979 ($\\sim$\\,JD\\,24\\,44000) only $V$ declined; see text.}\n  \\label{Fig6}\n  \\end{figure}\n\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{2019_FIG2fin_HR8752.pdf}}\n  \\caption{Similar to Fig.\\,6, but now for the LTV of $(B-V)$ and $(U-B)$ of HR\\,8752, 1942--1994. We note that after JD\\,24\\,36000 (1957) the star suddenly reddened during the next $\\sim$\\,16\\,y by $\\sim$\\,0\\fm5 until the eruption in 1973. The spectral study by Luck (1975) revealed a temperature of 4000\\,K at JD\\,24\\,41910, August\\,1973.   }\n   \\label{Fig7}\n \\end{figure}\n\n  \\begin{figure}\n   \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{2019_FIG1up_HD179821.pdf}}\n   \\caption{Photographic LTV ($B_{\\rm pg}$) of HD\\,179821 between 1899 and 1989 (Arkhipova et al. 2001). We note the deep minima between JD\\,24\\,24000 and JD\\,24\\,38000 ($\\sim$\\,1925 and $\\sim$\\,1960) indicating a very red and cool episode, favourable for an eruption; see text. HR\\,8752 showed the same type of photometric history before its blue loop evolution, in which an eruptive episode occurred (1973) according to spectroscopic observations (Figs.\\,6 and 7). }\n   \\label{Fig8}\n \\end{figure}\n\n\\begin{figure}\n   \\resizebox{\\hsize}{!}{\\includegraphics{img030.pdf}}\n  \\caption{Splendid coverage of the LTVs in $V$, $B$, and $U$ of HD\\,179821, between 1990 and 2009 (based on observations by Arkhipova et al. 2001, 2009; Le Coroller et al. 2003;\n  and Hrivnak et al. 2001). We note that dips and bumps in $V$, $B,$ and $U$ do not vary in concert. It appears that the LTVs in $JK$ of Arkhipova et al. 2009 and Hrivnak et al. 1989 (see Fig.\\,9 in Arkhipova et al. 2009) mainly mimic the $V$ trend, and more or less the bumps and dips between JD\\,24\\,49000 and JD\\,24\\,50500. We witness here the presumable evolution of HD\\,179821 along a blue loop in the direction of the Yellow Void (just like HR\\,8752 did after 1990) ending around JD\\,24\\,53000 (2004), after which HD\\,179821 moved back on a red loop, see text; see Fig.\\,10.}\n  \\label{Fig9}\n  \\end{figure}\n\n \\begin{figure}\n  \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{img032.pdf}}\n  \\caption{Same as Fig.\\,9 but for $(B-V)$ and $(U-B)$ of HD\\,179821 between 1990 and 2010. The bluest value reached during the blue loop evolution is $(B-V)$\\,=\\,1.25, thus, $(B-V)_{0}$\\,=\\,0.55 (extinctions in Table\\,1) happened at JD\\,24\\,53000 (2004). The mean spectroscopic temperature T(Sp) at the time according to Arkhipova et al. (2009) is 6800\\,K\\,$\\pm$\\,50\\,K. After 2004 the star moved back on a red loop, see text.}\n  \\label{Fig10}\n  \\end{figure}\n\n{\\bf --HD\\,179821:} The first record of an LTV of HD\\,179821 started in 1899 (JD\\,24\\,16000) with a photographic $B_{\\rm pg}$ dataset until 1989 (collected by Arkhipova et al. 2001).\nIndividual data points, or small concentrations of data points are connected by a line (Fig.\\,8). This LTV is of course less reliable between 1899 and 1989, yet it gives a good impression of the  instability of the star at the time and its cool phase.\n Unfortunately, no details of a possible eruption were observed, but we consider it not unlikely that one occurred somewhere between 1925 and 1960.\nOur assembled data show on average a gradual rise in light amplitude of $\\sim$\\,0\\fm5. There is some overlap with modern observations by means of two data points which precisely match the modern $B$ magnitude sequence made photoelectrically until about 1989 ($\\sim$\\,JD\\,24\\,48000, Figs.\\,9 and 10). Therefore, there is no significant reason to mistrust the reliability of the observations dating back to 1899. Thereafter, the photometric instability from the visual to the UV of the LTVs was very small: 0\\fm1--0\\fm2, until 2009 (JD\\,24\\,53000).\n\nIt appears that the $B_{\\rm pg}$ was 9.5 in 1899 ($\\sim$\\,JD\\,24\\,15000), declined to a deep minimum with $B_{\\rm pg}$\\,10 around 1925 ($\\sim$\\,JD\\,24\\,24000), and was at a second deep minimum around 1960 ($\\sim$\\,JD\\,24\\,37000). The $B$ in Fig.\\,9 gradually rose to a magnitude of 9.4--9.3 in 2009 ($\\sim$\\,JD\\,24\\,55000), showing small fluctuations, at most by $\\pm$\\,0\\fm1, just like in $V$.\n\nThe modern portion of the LTV in $U$ (Fig.\\,9) from JD\\,24\\,48000 until JD\\,24\\,52500 (1990--2003) shows that the star was still rising in brightness by 0\\fm4.\n\nThe gradual blueing of $(B-V)$ in Fig.\\,10 between JD\\,24\\,48000 (1990) and JD\\,24\\,53000 (2004) and the subsequent reddening until 2009  is of crucial importance for the interpretation of the evolutionary state of HD\\,179821 below. This reddening trend is supported by the new $UBV$ photometry until 2017 by Ikonnikova et al. (2018). If HD\\,179821 is indeed a YHG an eruptive episode can be expected within a few decades.\n\nNoteworthy are two bumps and dips in the LTVs for $V$, $B,$ and $U$ in Fig.\\,9 between JD\\,24\\,49000 and JD\\,24\\,50500. Sometimes $UB$ and $V$ run in opposite directions. The LTVs in $JK$ shown in Fig.\\,6 of Arkhipova et al. (2009) mimic the one in $V$. The same is more or less the case for the bumps and dips.\n\n\\begin{figure}\n   \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{2019_FIG1new3_RHOCAS.pdf}}\n   \\caption{Long-term variations in the brightness $UBVRI$ (Johnson system) of $\\rho$\\,Cas between 1965 and 2017. The arrows and the dates at the top mark eruptive episodes. Data points at JD\\,$\\sim$\\,24\\,58000 are observations by EJvB. Between the $V$ and $B$ light curves the mean quasi-periods are shown with their $\\sigma$ for some selected time\n  intervals. Their mean durations are represented by the length of the bars. The number of cycles used are bracketed.\n  A correlation between the trend of the mean quasi-periods and the $(B-V)$ is obvious: they increase and decrease when the $(B-V)$ becomes red and blue, respectively (while $V$ is almost constant, $B$ shows a cyclic fainting and brightening before and after an eruption).\n  The eight $B$-data points represented by circles are based on a $(V-R)$\/$(B-V)$ relation\n when no $B$ observations were made. }\n  \\label{Fig 11}\n  \\end{figure}\n\n \\begin{figure}\n   \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{2019_FIG2new2_RHOCAS.pdf}}\n    \\caption{Similar to Fig.\\,11 but for the five colour indices of the $UBVRI$ (Johnson system) of $\\rho$\\,Cas between 1965 and 2017. The $(B-V)$ data points\n  represented by circles are based on a $(V-R)$\/$(B-V)$ relation when no $B$ observations were made (Fig.\\,11).\n  The arrows and the dates at the top mark eruptive episodes. We note that eruptions preferably take place when the LTV of $(B-V)$ is relatively red, therefore when the star resides in the red part on the HR diagram.}\n    \\label{Fig 12}\n    \\end{figure}\n\n {\\bf --$\\rho$\\,Cas :} Figure\\,11 shows the LTVs of $\\rho$\\,Cas for the $UBVRI$ data between 1965 and 2015. The arrows at the top indicate observed eruptive episodes. The mean quasi-periods with standard deviations and the number of cycles used (bracketed) are given in the second panel. The selected time intervals are indicated by the length of the bar.\n Figure\\,12 shows the LTVs for the colour indices $(B-V)$, $(U-B)$, $(V-R),$ and $(V-I),$  the magnitude scale of which is about three times larger than in Fig.\\,11. The LTVs are based on the observations by many observers (Sect.\\,2.1.), but the majority are by Henry (1995, 1999). The preference of eruptions to occur when the star is relatively faint and red is obvious.\n\nThe correlation between the trends of the quasi-periods and the $(B-V)$ is obvious and depends on the type of the evolutionary track: increasing and decreasing during RL and BL evolutions, respectively (Figs.\\,11 and 12), just like HR\\,8752.\n\n{\\bf --HR\\,5171A :} For HR\\,5171A we had the disposition of photographic $B$ magnitudes (Johnson system) to reconstruct its photometric history from 1900 until 1950 and between 1978 and\n  1989. They were obtained from Harvard plates of which the scanning was made within the framework of the DASCH scanning project. The measurements were kindly put at\n our disposal by Dr. Josh Grindlay, head of this project, before the official release of the data.\n The position of HR\\,5171A was close to the edge of the plate. Probable errors are 0\\fm2--0\\fm3. It should be noted that the Harvard magnitudes include the nearby blue optical\n  companion HR\\,5171B (10$\\arcsec$), which is likely not a physical companion, but a member of Gum48d, contrary to HR\\,5171A, at least if its proposed shorter distance in Sect.\\,2.9. is correct. Its contribution to the total brightness is, depending on the brightness of HR\\,5171A, smaller than $\\sim$~0\\fm25.\n  According to Stickland \\& Harmer (1978), HR\\,8752  also has a nearby B-type companion, but that one is a physical companion; see also Nieuwenhuijzen et al. (2012, Sect.\\,1.1.\n\n  The brightness limits ($B$\\,=\\,8\\fm4--9\\fm8) and the mean magnitude (9\\fm1) of the data points in the 1900--1950 set which is not shown in the present paper appear to be similar to the modern ones after 1952 (Fig.\\,1 in Paper\\,I, and Fig.\\, 3 in Chesneau et al. 2014). These papers refer to the original sources like those of Harvey 1972; Humphreys et al. 1971; Dean 1980; from ASAS-3 by Pojmanski 2002; from AAVSO amongst others by Otero; and from the Long-Term Photometry of Variables (LTPV) group: Sterken 1983, Manfroid et al. 1991, Sterken et al. 1993; from Hipparcos discussed by van Leeuwen et al. 1998. The Harvard $B$ data collected in the time interval 1978--1989 match the photoelectric data satisfactory. Our conclusion is that HR\\,5171A did not alter its variability pattern during the last 120\\,yrs.\n\n  Figures\\,13 and 14 represent the LTVs of HR\\,5171A in $B$ and $V$, and in $(B-V)$, respectively, between 1953 and 2018: the black lines connecting the black dots. These latter figures are mainly based on the papers by van Genderen (1979 and 1992 Fig.\\,1) and Chesneau et al. (2014) discussing the photometric history of HR\\,5171A. The meaning of the other curves in Fig.\\,14 is explained in Sect.\\,3.4.\n\n  Between 1994 and 2012 observations were made only by visual observers, but the means, represented by the dashed oscillating curves fit the sketched LTV in $V$ reasonably well. However, some systematic deviations can be attributed to the fact that for the construction of part of the LTV (between JD\\,24\\,50900 and JD\\,24\\,55500), we relied mostly on those made by Otero (see Chesneau et al. 2014) and on the All Sky Automated Survey-3 (ASAS) data (Pojmanski 2002; see Fig.\\,3 in Chesneau et al. 2014). Systematic differences in the magnitude scales of different observers amounting to a few 0\\fm1 are normal. The last set of $BV$ photometry (2013--2018) is listed in \\,Table\\,I.1 of the Appendix.\n\n  Only three mean quasi-periods could be derived (top of Fig.\\,13), as reliable pulsations between 1995 and 2012 were lacking. As a consequence of the above, and the fact that after the 1975 eruption the star became heavily obscured (discussed in Sect.\\,3.4.1.) no reliable correlation between mean quasi-periods and colour could be determined. On the contrary, the third mean quasi-period on the very right has become significantly shorter which is in agreement with the observed blueing trend until the last observations of 2018.\n\n The 1975 eruption named after Dean (1980) was only observed during the brightness rise after a deep minimum. The second eruption of HR\\,5171A, including the deep minimum, was well observed, but only in the visual: the Otero minimum of 2000. The maximum just before the deep decline is the eruption maximum. Alas, multi-colour photometry and spectroscopy were lacking.\n\n The arrow at JD\\,24\\,47300, 1988 (see Fig.\\,13), is a local minimum of the LTV during the $VBLUW$ photometric time series (Fig.\\,3 upper panel), containing three pulsation cycles nos.\\,15--17 (in Fig.\\,1 of Paper\\,I).\nThe deepest visual pulsation minimum is called the Jones-Williams minimum of 1995 (JD\\,24\\,49750, Fig.\\,3 upper panel).\n The smoothed LTV in the $VBLUW$ system of Walraven is shown and discussed in\\,Fig.\\,J.1 of the Appendix.\n\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90] {img022.pdf}}\n  \\caption{Long-term variables for $V$ and $B$ of HR\\,5171A 1953--2018. Time intervals (with mean quasi-periods and the sigma) are indicated by bars at the top. Bracketed numbers are the number of cycles. The mean quasi-period (494$^{\\rm d}$) between 1970 and 1993 is based on a least-squares solution by van Leeuwen et al. (1998). See text for a discussion on the mean quasi-periods after $\\sim$\\,2000 (Sect.\\,3.1.).\n  The dashed light curve is based on the visual AAVSO data. The Otero-minimum is part of the massive 2000 eruption. Local brightness minima (of LTVs), or of individual pulsations are: the $VBLUW$ minimum (JD\\,24\\,47300), the Jones--Williams minimum (the deepest one at the time, JD\\,24\\,49750), and the Blown-minimum at JD\\,24\\,57330. The last two minima are identifiable in the  upper and lower panels of Fig.\\,3, respectively. The squares on the very right are individual $BV$ observations made by GDS between 2013 and 2018 (Appendix\\,Table\\,I.1.); see Fig.\\,15 for a detailed visual light curve in the same time interval. In this time interval ($\\sim$\\,JD\\,24\\,56755, 2014) the temperature was determined by Wittkovski et al. (2017a): 4290\\,K\\,$\\pm$\\,760\\,K. }\n  \\label{Fig 13}\n  \\end{figure}\n\n\n \\begin{figure}\n  \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{img025.pdf}}\n   \\caption{Similar to Fig.\\,13, but for the observed LTV of $(B-V)$ of HR\\,5171A, 1953--2018: the black dots connected by black lines.  We note the long-lasting reddening episode between $\\sim$\\,JD\\,24\\,37000 (1960) and $\\sim$\\,JD\\,24\\,44500 (1981).\n  All other curves are only schematic. In red and blue: the $(B-V)$ variations due to the RL and BL evolutions, respectively; black dotted peaks: the Dean 1975  and Otero 2000 eruptions; black dotted curve below and on the very right: the supposed blueing trend;\n  black dashed curve running steeply downward: the supposed reddening by a shell ejection after the 1975 eruption. The result of the latter and the BL evolution above is the observed reddening of the LTV: the black dots connected by black lines between the 1975 eruption and $\\sim$\\,JD\\,24\\,46000.\n  In the time interval between JD\\,24\\,50000 and JD\\,24\\,56000 (1994--2013) only visual observations were made. The small dataset (squares) on the very right (2013--2018) represents individual photoelectric observations by GDS (\\,Table\\,I.1 of the Appendix.); see  Fig.\\,15 for the detailed visual light curve at the time.    }\n    \\label{Fig 14}\n   \\end{figure}\n\n\n\\subsection{Brightness and colour ranges of the red and blue 'evolutionary modes'.}\n Table\\,J.1 of the Appendix presents a summary of the maximum ranges of the LTVs in brightness and colour index. These ranges serve a useful purpose, amongst others, giving a more precise determination of the actual evolutionary trends on the HR diagram in a later paper. Those of HD\\,179821 are smallest. The largest ranges for the four objects occur in $U$ and $B$, that is, up to 1\\fm2, and the smallest ones occur in $R$ and $I$. Those in $(U-B)$ are larger than in $(B-V)$.\n\n Another property is that the trends of the light variations in different pass bands often run in variable directions, and therefore in various `modes'. For example, $V$ and $B$ run in opposite directions and with different rates for a few years, then both decline in brightness with equal rates for another couple of years, after which $V$ declines much faster than the rise in $B$, and so on.\n\n We determined the evolutionary modes for HR\\,8752, $\\rho$\\,Cas, and HD\\,179821 and listed them chronologically in Table\\,4, concentrating on those in $V$ and $B$ only, as $(B-V)$ is (apart from the reddening effect by the high-opacity layer) a temperature indicator. The red and blue evolutionary loops of HR\\,8752 are shown in Fig.\\,6 in Nieuwenhuijzen et al. 2012. The modes for HR\\,5171A are collected in Table\\,5, among which there are a few short-term modes.\n\n These tables list the type of modes with respect to the trends of $V$ and $B$ (brightness up, down, or constant), and of $(B-V)$ (to the red or to the blue, the latter in boldface), and the dates, JDs, and their durations.\n\n\n \\begin{table*}\n    \\caption{Red and blue evolutionary modes for the LTVs of HR\\,8752, $\\rho$\\,Cas, and HD\\,179821 listed in chronological order. Second column: u\\,=\\,up\\,=\\, brightness increasing, d\\,=\\,down\\,=\\,brightness\n  declining, if\n  double, i.e. uu and dd, the increasing and declining, respectively, are stronger than in the other channel, c\\,=\\, constant. Third column:\n  const.\\,= \\,constant. For clarity, the trends `blue' are in boldface. For HR\\,8752 and $\\rho$\\,Cas, the transition from a red to a blue trend is caused by an eruption (1973, and 1986, 2000, 2013, respectively). Such a significant event for HD\\,179821 should have happened, but was obviously missed. }\n    \\label{table:4}\n    \\centering\n   \\begin{tabular}{l l c c c c}\n  \\hline\\hline\n   Star  &  Mode &            Trend     & Dates & JD-      & Duration \\\\\n         &                 & of $(B-V)$ &       & 24\\,00000    & (d)     \\\\\n  \\hline\n   HR\\,8752    & $B_{\\rm pg}$-d &         & 1899--1923 & 15000-23600 & 8600  \\\\\n               & $B_{\\rm pg}$-u &         & 1923--2001 & 23600-52000 & 28400 \\\\\n               & $V$-uu B-u   & red  &  1942--1950 & 30500--33500 & 3000  \\\\\n               & $V$-d $B$-d  & const & 1950--1957 & 33500--36000 & 2509 \\\\\n               & $V$-d $B$-d  & red   & 1957--1976 & 36000--43000 & 7000  \\\\\n               & $V$-d $B$-u  & {\\bf blue} & 1976--2005 & 43000--53500 & 10500  \\\\\n   \\hline\n   $\\rho$\\,Cas & $V$-u $B$-d  & red & 1965--1982 & 34000--45000 & 6000  \\\\\n               & $V$-d $B$-dd & red & 1982--1986 & 45000--46500 & 1500  \\\\\n               & $V$-u $B$-uu & {\\bf blue} & 1986--1991 & 46500--48500 & 2000 \\\\\n               & $V$-c $B$-d & red  & 1991--1998 & 48500--51000  & 2500 \\\\\n               & $V$-d $B$-dd & red & 1998--2000 & 51000--51300  & 300  \\\\\n               & $V$-c $B$-uu & {\\bf blue} & 2001--2009 & 52000--55500  & 3500  \\\\\n               & $V$-c $B$-dd & red  & 2009--2013 & 55000--56500 & 1500 \\\\\n               & $V$-c $B$-u  & {\\bf blue} & 2013--2015 & 56500--58100  & 1600  \\\\\n  \\hline\n   HD\\,179821  & $V$-d $B$-u & {\\bf blue}  & 1990--1993 & 48000--49200  & 1200  \\\\\n               & $V$-dd $B$-uu & {\\bf blue} & 1993--2004 & 49200--53200 & 4000  \\\\\n               & $V$-c $B$-d  & red &  2004--2008 & 53200--54500  & 1300  \\\\\n  \\hline\n  \\end{tabular}\n  \\end{table*}\n\n\\begin{table*}\n \\caption{Similar to Table\\,4, but now for HR\\,5171A. The transitions between blue and red modes between 1973 and 1977, just before and after the 1975 eruption, are amazingly short: 200$^{\\rm d}$--500$^{\\rm d}$ compared to the other ones, and those of HR\\,8752, $\\rho$\\,Cas, and HD\\,179821 in Table\\,4; see Sect.\\,3.4.1. for a plausible cause. We note the gap between 1995 and 2013 which is caused by the lack of $BV$ photometry.\n The $BV$ data between 2013 and 2018 are listed in \\,Table\\,I.1 of the Appendix. The 2013--2018 mode is not yet finished, and therefore the duration is a lower limit   }\n  \\label{table:5}\n  \\centering\n   \\begin{tabular}{l c c c c }\n   \\hline\\hline\n   Mode & Trend      & Dates & JD-   & Duration  \\\\\n        & of $(B-V)$ &       & 24\\,00000   & (d)  \\\\\n  \\hline\n  $V$-c $B$-c & const & 1953--1958 & 34400--36400 & 2000 \\\\\n  $V$-c $B$-d & red   & 1958--1962 & 36400--37800 & 1400  \\\\\n  $V$-d $B$-dd & red  & 1962--1965 & 37800--38800 & 1000  \\\\\n  $V$-d $B$-dd & red  & 1965--1969 & 38800--40400 & 1600  \\\\\n  $V$-d $B$-dd & red  & 1969--1973 & 40400--42000 & 1600  \\\\\n ($V$-dd $B$-d) & {\\bf blue} & 1973--1975 & 42000--42500 & 500 \\\\\n  $V$-uu $B$-u & red  & 1975--1976 & 42500--43000 & 500  \\\\\n  $V$-u $B$-uu & {\\bf blue} & 1976--1977 & 43000--43200 & 200  \\\\\n  $V$-c $B$-d & red   & 1977--1981 & 43200--44800 & 1600  \\\\\n  $V$-c $B$-c & const & 1981--1985 & 44800--46100 & 1300 \\\\\n  $V$-d $B$-d & const & 1985--1988 & 46100--47300 & 1200  \\\\\n  $V$-uu $B$-u & red  & 1988--1991 & 47300--48400 & 1100  \\\\\n  $V$-dd $B$-d & {\\bf blue} & 1991--1995 & 48400--50000 & 1600 \\\\\n\n  $V$-dd $B$-d   & {\\bf blue} & 2013--2018   & 56330--58140 & 1810 \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Red and blue loop evolutions: their $(B-V)$ ranges and quasi-period trends.   }\n\nBelow we derive approximate evolutionary tracks, and an important quantity: the range in $(B-V)$ during the evolution, and thus the variation of T$_{\\rm eff}$, assuming that it is almost independent of the atmospheric absorption (Sects.\\,2.4.--2.6.). Additionally, we try to establish the true nature of HD\\,179821 and discuss its probable isolated position in space.\n\nThe study of HR\\,8752 by Nieuwenhuijzen et al. (2012) revealed that HR\\,8752 evolved along a red loop evolution and after the 1973 eruption along a blue loop evolution (hereafter called RL and BL evolutions). Therefore, all red modes and all blue modes listed in Tables\\,4 and 5 and preceding and following an eruption represent shorter pieces of one RL and one BL evolution, respectively.\nThis probably means that the evolution tracks on the HR diagram are not always straight, but show irregularities.\nBelow,  we firstly discuss these RL and BL evolutions of HR\\,8752, $\\rho$\\,Cas, and HD\\,179821.\n\n{\\bf --HR\\,8752:} The construction of a RL from 1850 (best documented from 1895) until the 1973 eruption and subsequently the BL until 2005, are based on a mix of different observational techniques (Nieuwenhuijzen et al. 2014, their Table\\,4 and Figs.\\,5 to 24).\nThe $(B-V)$ ranges are 0\\fm16 (1895--1963) and 0\\fm64 (1976--2005). Between 1963 and 1976, no $BV$ photometry was available.\n\nObserved and model timescales (by Meynet et al. 1994), that is, a few decades, are of the same order. Multi-colour photometry has been absent since 1993, but spectral observations were made until between 2000 and 2005 from which T$_{\\rm eff}$ and log\\,g from Kurucz's LTE models could be derived (Nieuwenhuijzen et al. 2012, their Table\\,4 and Fig.\\,2). During this BL evolution, temperatures and gravities fit the extrapolation line of the gradual rise of T$_{\\rm eff}$ up to 8000\\,K and log\\,g from 1993 until 2005 very well.\n\nNieuwenhuijzen et al. (2012) claimed that HR\\,8752 left the first instability region on the HR diagram, crossed the Yellow Void (Fig.\\,1), and then was on its way to stability further to the blue, and that during further evolution it must still go through the second potential unstable region, where He starts to ionize. However, as mentioned before, the recent $BV$ photometry (by EJvB, Appendix\\,B:2.1.) indicates that in 2017\/2019 the $(B-V)_{0}$ of the star stopped blueing, but is still in the Yellow-Blue Void (Fig.\\,1), and shows some variability in brightness and colour owing to its very-low-amplitude pulsations.\nAn additional analysis made by us of the AAVSO $V$ (Johnson) observations of HR\\,8752 from 1993 until 2019 is presented in Appendix\\,K:\\,3.3.\n\n{\\bf --$\\rho$\\,Cas :} According to our collection of fragments of LTVs from numerous papers, $\\rho$\\,Cas has passed through three RL and two BL evolutions since the $BV$ photometry started in 1968; see the $(B-V)$ curve in Fig.\\,12: a RL from 1968 to the 1986 eruption triggering a BL until about 1991, succeeded by a RL until the 2000 eruption, triggering the next BL until about 2008, then a RL until the end of the figure in 2018. Also, here the timescales are one to a few decades only.\nThe ranges in $(B-V)$ of two RL and two BL evolutions lie between 0\\fm20 and 0\\fm45, meaning that the temperature ranges can be derived after correction for reddening (Table\\,1). They lie in the range of 5200\\,K to 4500\\,K (excluding the extreme temperatures during the eruption maxima and deep minima of about 7500\\,K and 4000\\,K, respectively).\n\n Based on the existence of such a `sequence of events', likely valid for all YHGs, we conclude that a third blue loop of $\\rho$\\,Cas must have taken place after the massive 1946 eruption (light curve in visual and photographic magnitudes by Gaposhkin 1949). Indeed, one to two decades later, in the 1960s and 1970s (in Figs.\\,11 and 12 on the very left), the spectral type was F8, which is relatively blue (Humphreys 1978; de Jager et al.1988; Arellano Ferro \\& Mendoza 1993). The LTV at the time $(B-V)_{0}$\\,=\\,0.60, which corresponds with T$_{\\rm eff}$\\,$\\sim$\\,6000\\,K.\n\n According to Fig.\\,12, the blue loop after the 2000 eruption reached roughly the same $(B-V)_{0}$ and consequently the same temperature as the eruption of 1946 until about 1968 (the latter date on the very left of Fig.\\,12).\n Whether this was also the case for the blue loop after the 1986 eruption is uncertain, because of some inconsistency between the $(B-V)$ based on $BV$ photometry (dots in Fig.\\,12) and the $(B-V)$ based on a transformation from the $VRI$ photometry (circles in Fig.\\,12). It appears that an inconsistency exists between the trends of dots and circles which is disappointing, but usually the transformation yields more satisfactory results.\n\n{\\bf --HD\\,179821 :} One glance at the light curves of this object was enough to recognize a BL evolution. It is staggering how closely the past history of HD\\,179821 imitated that of HR\\,8752, even if one compares the pulsation properties until 2018 made with the aid of the AAVSO Light-Curve Generator. The amplitudes and timescales of the quasi-periods are smaller and shorter (100$^{\\rm d}$--150$^{\\rm d}$), respectively, as far as the visual observations of HR\\,8752 allow the comparison with the $UBV$ photometry by Arkhipova et al (2001, 2009), Le Coroller et al. (2003) of HD\\,179821. The pulsations of both objects also share some photometric morphological properties.\nIkonnikova et al. (2018) reported that the quasi-periods of HD\\,179821 became longer than 250\\,d between 2010 and 2017, which can be expected when evolving along an RL evolution!\n\nThe BL of HD\\,179821 started somewhere in the time interval 1925--1960, because at the time the star was very faint in B$_{\\rm pg}$, and therefore very cool; see Fig.\\,8.\nThe blue loop ended in 2004 (about JD\\,24\\,53000) when $(B-V)$ reddened again, marking a new RL evolution (Figs.\\,9 and 10). The range of the colour $(B-V)$\\,=\\,0\\fm22.\n\nGiven the characteristic `cyclic sequence of events' for HR\\,8752 and HD\\,179821, an eruption could have happened somewhere between 1925 and 1960, but like in the case of HR\\,8752 (Sect.\\,3.1.) stayed unnoticed.\nThe timescale is uncertain, but is approximately five decades, and therefore not in disagreement with theory\n(the gaps in the time series of Fig.\\,8 are in most cases much longer than the duration of eruptions: 400$^{\\rm d}$--700$^{\\rm d}$).\n\nThe bluest colour (corrected for IS extinction) of HD\\,179821 in 2004 was $(B-V)_{0}$\\,$\\sim$\\,0\\fm55, its reddest colour in 1990, 0\\fm77. Hence, the maximum photometric temperature is T(Phot)\\,=\\,5290\\,K (dJN), while at the time the mean spectroscopic temperature according to Arkhipova et al. (2009) was T(Sp)= 6800\\,K\\,$\\pm$\\,50\\,K (a difference of 1500\\,K, see Sects.\\,2.4.-- 2.6.). According to its position at the time in Fig.\\,1, it stopped just before entering the Yellow-Blue Void and was heading back according to the available observations until 2009 (Fig.\\,10).\nWith the Straizhys (1982) temperature calibration T(Phot) a comparable result was obtained by Arkhipova et al. (2009): 5400--6000\\,K.\n(As noted before new $UBV$ photometry by Ikonnikova et al. (2018), HD\\,179821 continued its reddening trend until the end of the observations in 2017).\n\nAll the similarities above suggests that HD\\,179821 is not a RSG, or a post-AGB star as is often claimed, but presumably a YHG at a very large distance (M$_{\\rm bol}$\\,$<$\\,-8), supporting conclusions by Gledhill et al. 2002, Gledhill \\& Takami 2001, Molster et al. 2002,  Arkhipova et al. (2009), and Oudmaijer et al. (2009). However, as a consequence, the large z-distance to the galactic plane creates kinematical problems, which can only be solved by a reliable trigonometric parallax (Arkhipova et al. 2009). Its large distance and its isolated position with respect to its place of birth (a young stellar cluster) is not unusual: see Smith \\& Tombleson (2015: LBVs are `anti-social'). The isolated positions of many LBVs (the probable descendants of YHGs) are remarkable. Smith \\& Tombleson believe that LBVs are flung out of their former binary orbit by a SN explosion. See also the related discussions by Humphreys et al. (2016) and Aghakhanloo et al. (2017). In the case of the suggested distance reduction for HR\\,5171A, as discussed in Sect.\\,2.9., its position also becomes isolated, as it will no longer be a member of the young stellar cluster Gum48d.\n\nOur conclusions are that the observed $(B-V)$ ranges for the BL evolution lie roughly between 0\\fm2 and 0\\fm6, with T$_{\\rm eff}$ roughly between 8000\\,K and 4000\\,K, roughly corrected for the too red $(B-V)$ values (Table\\,A.1 of the Appendix).\nOther novelties are that HD\\,179821 is likely a YHG, and that all four YHGs seem to be isolated objects.\n\n\\subsection{HR\\,5171A - a special YHG}\n\n\\subsubsection{The mysterious reddening episode 1960--1981.}\n\nThe most confusing property of HR\\,5171A for a generation of researchers was the exceptional reddening episode by 0\\fm7 in $(B-V)$ observed between about 1960 ($\\sim$\\,JD\\,24\\,36000) and 1981 ($\\sim$\\,JD\\,24\\,44000); see Figs.\\,13 and 14. This happened before and after the 1975 eruption. After 1981 the reddening increase came to an end. Below we offer a plausible explanation with the aid of Fig.\\,14.\n(Its caption explains the meaning of the other curves: only schematically sketched).\n\nFirstly, the long-lasting steep reddening episode starting around 1960 until 1981 cannot be caused by an increasing dust extinction, as the $B$ decline relative to that of $V$ happened much too quickly.\nSecondly, because of the quick succession of three short-lasting modes: 500$^{\\rm d}$, 200$^{\\rm d}$ and 500$^{\\rm d}$), a blue, a red, and a blue one, respectively, between 1973 and 1977 (Table\\,5). Obviously, something exceptional happened at the time, disturbing the brightening phase after the 1975 eruption, as well as the two colour variations of the following pulsations in 1977 and 1978 (numbered 7 and 8 in Fig.\\,1 in Paper\\,I). As observations were scarce at the time, no reliable details can be offered.\n\nHowever, as the three other YHGs highlight an alternating succession of a RL, an eruption, and a BL, with a timescale of a few decades only, this should also apply to HR\\,5171A, but we are convinced that another event should have happened simultaneously.\n\nWe propose the following plausible scenario. The sketched alternating sequence of red and blue curves in Fig.\\,14 represents the simplified and inevitable representation of the RL and BL evolutions, bounded to the two eruptions. We believe that the first observed reddening episode 1960--1974 was in fact a RL evolution, represented by the first red dashed curve, as it has a $(B-V)$ range of 0\\fm35 (falling within the range of the other three YHGs, Sect.\\,3.3.). Additionally, the $B$ decline relative to the decline of $V$ 1960--1974 in Fig.\\,13 is quite unlike an increasing extinction by more dust.\n\nThe reddening in the second episode 1976--1981 started right after the 1975 eruption; it increased even faster than the one between 1960--1974, and stopped in 1981. This is quite awkward. It is plausible that the latter reddening episode was mainly due to a massive shell ejection during the 1975 eruption, schematically represented by the black dashed curve in Fig.\\,14. The $B$ absorption was obviously much higher than in $V$. This event happened simultaneously with the first BL evolution, starting right after the 1975 eruption. The result of both events produced the observed LTV represented by the black dots connected by straight lines.\n\nA shell event happening right after the 1975 eruption is likely considering the existence of a massive optically thick extended molecular and dust envelope around HR\\,5171A; see Sects.\\,2.9 and 3.4.2. Obviously, such events happened in the past as well.\n\nThis first BL evolution was succeeded by a new (second) RL evolution bounded by the impending eruption in 2000. Hence, the observed $(B-V)$ in Fig.\\,14 became bluer as $V$ declined more than $B$ between 1990 and 1995; see Fig.\\,13. We note that this probably explains the relatively large depth of the Jones--Williams pulsation minimum at JD\\,24\\,9750 (1995); see Figs.\\,3, and 13.\n\nThe continuation of the dashed black curve to the right at the bottom of Fig.\\,14 roughly indicates the presumable continuation of the intrinsic reddening by the shell, and its subsequent decrease. After the 2000 eruption, a second BL evolution then developed which is responsible for the blueing trend of $(B-V)$ after 2001, and was still going on early 2019 (Sect.\\,3.4.3.).\nIn conclusion, here we offer a new and very plausible explanation for the persistent reddening of HR\\,5171A between 1960 and 1981.\n\n\\begin{figure}\n \\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{img033.pdf}}\n  \\caption{Visual AAVSO light curve 2011--2018 (red lines based on 50$^{\\rm }$ d averages) of HR\\,5171A showing the last normal pulsation with the Blown-minimum (red arrow at $\\sim$\\,JD\\,24\\,57330, early October 2015, cf.  second\npanel of Fig.\\,3), during an observed portion of the BL evolution showing sudden decreasing quasi-periods and declining light amplitudes. The behaviour after JD\\,24\\,57650 is similar to those exhibited by HR\\,8752 and HD\\,179821 during their BLs. The green dots are the photo-electric $V$ observations from Table\\,I.1, which are brighter than the visual estimates by 0\\fm2--0\\fm4 as expected. The visual ones should be shifted upwards, but for clarity this was not done. See text for further explanation.  }\n   \\label{Fig.15}\n  \\end{figure}\n\n\n\\subsubsection{The 1977 excess of Balmer continuum radiation of HR\\,5171A and the near-infrared $JHKL$ connection. }\n\nFirstly, we relate the Balmer continuum excess of HR\\,5171A observed in 1977 to its 1975 eruption episode and shell event with the aid of the $VBLUW$ photometric system. Subsequently, the effects of RL and BL evolutions and the 1975 eruption on the $JHKL$ light curves are discussed.\n\nOne can only speculate as to any relation between the matter blown into space after the atmospheric explosion of 1975 and the anomalous high UV excess happening during the development of the very-high-amplitude pulsation of 1977 immediately following the eruption (labelled no.\\,7 in Fig.\\,1 of Paper\\,I). The Balmer continuum radiation excess amounted to about 1$^{\\rm m}$ (see Fig.\\,1 in van Genderen et al.\\,2015). No such excess was observed in 1971, 1973, or between 1980 and 1991 when also $VBLUW$ photometry was made.\nA glance at the $B_{\\rm pg}$ light curve of the Harvard DASCH scanning project (not shown, only discussed in Sect.\\,3.1.) indicates that such deep brightness declines could have happened in the past as well. (Maximum light was not observed during the 1975 eruptive episode, nor was the deep brightness decline to a deep minimum; only the steep brightness rise was seen thereafter: see Fig.\\,13.).\n\nNevertheless, after 1981 the observed $(B-V)$ curve until the end of our time series in 2019 remained overly red by approximately 0\\fm6  compared to simplified RL and BL evolutions, of which the brightness level in Fig.\\,14 should be more or less correct. A causal connection with the 1975\\,\\,eruption and the effects of its abundant mass-loss is plausible. It is not surprising that HR\\,5171A is imbedded in an extended dust and molecular envelope (Chesneau et al. 2014; Wittkovski et al. 2017a, and Sect.\\,2.9). We note that an overall long-term variation of the Balmer continuum radiation exists (Appendix\\,J:3.4.2. and Fig.\\,J.1.).\n\nThe NIR $JHKL$ photometric light and colour curves, obtained at the SAAO and presented and discussed by Chesneau et al. (2014) in their Figs.\\,4 and 7, clearly show the same BL and RL evolutions after the 1975 eruption until 2012, just like the ones sketched in our Fig.\\,14. The first BL evolution reached maximum brightness at 1990, just like in our sketch of Fig.\\,14 at the bluest $(B-V)$. The 2000 eruption was reached after a RL with a brightness decline of 0\\fm6 in $L$ and $K$, but less in $H$ and $J$.\nThe NIR 1975 eruption minimum was at least 0\\fm3 fainter on average than that of the 2000 eruption. It is a pity that at the time of the latter, no $B$ magnitudes were made, meaning that the $(B-V)$ trend until 2012 is unknown (Fig.\\,14).\nThe trend of the colour indices was discussed by Chesneau et al. (2014: Sect.\\,4.1.).\n\nIn conclusion: HR\\,5171A was likely subject to a massive shell event after the 1975 eruption.  Blue- and red-loop evolutions appear to also be detectable in the NIR photometry.\n\n\\subsubsection{Current BL evolution of HR\\,5171A.}\n\nIn the autumn of 2015, our attention was aroused by the frequent reports of one of the present authors (EB) on the behaviour of HR\\,5171A based on his monitoring of the visual brightness and supported by many AAVSO colleagues. They are Alexandre Amorim, Brazil; Giorgio Di Scala, Australia; Hiroshi Matsuyama, Japan; Andrew Pearce, Australia; Peter Reinhard, Austria; Eric Blown, New Zealand; Eduardo Goncalves, Brazil; Alan Plummer, Australia; Antonio Padilla Filho, Brazil; Peter Williams, Australia: the visual brightness decline of a pulsation to a minimum in the autumn of 2015 seemed to happen relatively quickly (called the Blown-minimum at JD\\,24\\,57300, October 2015); see red arrow.\nThe pulsation amplitudes became suddenly smaller after 250 d (JD\\,24\\,57520), and the quasi-periods shorter until the end of our dataset at JD\\,24\\,58200 (March 2018), a time interval of about 2\\,yr. At the same time $(B-V)$, based on $BV$ photometry shown in \\,Table\\,I.1 of the Appendix, became bluer -- a similar behaviour to that of HR\\,8752 and $\\rho$\\,Cas.\n\nIt appeared that from the end of December 2015 to 2018, the light curve plotted by the AAVSO Light Curve Generator (JD\\,24\\,57100--JD\\,24\\,58300, March 2015--July 2018), shown in Fig.\\,15, revealed a most astonishing pattern: a mean decrease of the visual brightness (see Fig.\\,3 second panel of which Fig\\,15 represents the continuation), smaller light amplitudes, and much shorter quasi-periods within a time span of only $\\sim$\\,250\\,d. This happened between the maximum at JD\\,24\\,57400, right after the Blown-minimum, until JD\\,24\\,57650. During this pulsation, including the Blown-minimum and the maximum afterwards, some spectra were taken. An echelle spectrum was obtained, including H$\\alpha$ by E. Budding (ed.budding@gmail.com), and a high resolution spectrum was obtained by Dr. N.I. Morrell (nmorrell@lco.cl), but there were no abnormal features; the spectra looked normal (private comm. 2015).\n\nThe three bars at the bottom of Fig.\\,15 mark the epochs for the three $H$-band images of HR\\,5171A made by Wittkovski et al. (2017c). These images show a variable circumference, we presume not of the photosphere, but of its NIR $H$ band-radiating layer (although we expect the photosphere to be disfigured as well).\n\nIn conclusion, thanks to the continuous visual observations, we detected a surprising feature:\nduring the BL evolution in 2016, the pulsation amplitudes of HR\\,5171A suddenly appeared much smaller (and probably also with shorter periods). This phenomenon has not been observed before.\n\n\\begin{figure}\n   \\resizebox{\\hsize}{!}{\\includegraphics{FIG7_HD179821.pdf}}\n   \\caption{Demonstration of how T$_{\\rm eff}$(Phot) and R\/R$_{\\rm \\odot}$ of the light maxima of HD\\,179821 depend on the brightness and colour of the blue loop evolution. The stellar parameters of photometric temperature and radius variations were calculated for two possible distances 1\\,kpc (x signs) and 6\\,kpc (dots) using the dJN method. Brackets indicate uncertain results by a lack of data points in the light curves, encircled symbols represent the pulsation maximum at\n   JD\\,24\\,48550 of which $V$ is very bright, but the colour indices $(B-V)$ and especially the $(U-B)$ extremely red. The symbols labelled $V$ and $B$ refer to the maximum of a very peculiar pulsation at $\\sim$\\,JD\\,24\\,49300, of which the BL evolution in $V$ declined significantly sooner than in $B$. As most light maxima belong to a blue loop evolution (between JD\\, 24\\,48000 and JD\\,24\\,53000), Figs.\\,9 and 10, the photometric temperatures (lower panel) increase in concert with decreasing radii, and a declining visual brightness (upper panel).}\n \\label{Fig. 16}\n  \\end{figure}\n\n\\subsection{ Radius variation and contraction rate of HD\\,179821 during the BL evolution.}\n\nBelow we explore the effects of a BL evolution on the radius, the T(Phot), and the contraction rate, which has never been done before.\n\nAs we are dealing with evolutionary tracks on the HR diagram, we take it for granted that the stellar property variations keep pace with the temperature. Hence, one expects that radial velocity curves also show LTVs. Indeed, radial velocity monitoring campaigns of $\\rho$\\,Cas (monitored by Lobel et al. 2003, Figs.\\,1 and 4) reveal LTVs as well, and as far as we can judge these are more or less in concert with those of the visual brightness.\n\nAs long-term radial velocities for HD\\,179821 are lacking, we explored the effects of the BL evolution on its radii and photometric temperatures. We note that a calculated T(Phot) is always lower than a T(Sp). For example, an overly red $(B-V)$ is responsible for an overly low temperature, and according to the formula log\\,R\/R$_{\\rm \\odot}$\\,=\\,8.48--0.2\\,M$_{\\rm bol}$--2\\,log\\,T$_{\\rm eff}$, the radius becomes larger for an almost constant M$_{\\rm bol}$. Therefore, the results should not be taken too literally, as we are only interested in the general effects of a BL evolution on the stellar properties.\n\nThe illuminating result is presented in Fig.\\,16. The distance of HD\\,179821 is a serious matter of debate: between $\\sim$\\,1\\,kpc and $\\sim$\\,6\\,kpc (e.g. Surendinanath et al. 2002; Arkhipova et al. 2001, 2009; Le Coroller et al. 2003; Nordhaus et al. 2008; Patel et al. 2008; Oudmaijer et al. 2009; Ferguson \\& Ueta 2010; Sahin et al. 2016), M$_{\\rm bol}$\\,$\\sim$\\,-4.3 and -8.2, respectively, but according to our conclusion in Sect.\\,3.3., a large distance is favoured.\n\n\nWe calculated the trend of R\/R$_{\\rm \\odot}$ as a function of the $V$ and $(B-V)$ of the LTVs (here\nonly a blue loop track, of which the photometry is depicted in Figs.\\,9 and\n10, respectively) for most of the pulsation maxima of HD\\,179821, and for both distances. For the dJN method we then need to differentiate between an Ib--II supergiant and a hypergiant between the Ia$^{+}$ and Ia luminosity classes, for the short and long distances, respectively.\nAverages of the s-parameter for the two pairs of luminosity classes were used to derive T$_{\\rm eff}$ and BC (Appendix\\,F:2.7.). The extinction and reddening used are from Table\\,1.\nOn the left (dots) the trend of the radii for 6\\,kpc on the right (crosses) for 1\\,kpc.\nThe scatter around the mean lines is largely intrinsic. This can be explained as follows. About 30\\% of the light ($V$) and colour ($(B-V)$) curves of HD\\,179821 do not run in phase. As a consequence, at a fixed $V$ brightness, a great variety of $(B-V)$ colours exist, and therefore also a great variety of temperatures. According to the formula above, this causes a scatter as well in the computed radii, while the effect of the variation of M$_{\\rm bol}$ is small.\n\nThe radii at maximum light vary from 400 to 450\\,R$_{\\rm \\odot}$ and 60 to 80\\,R$_{\\rm \\odot}$ for 6 and 1 \\,kpc, respectively.\n The trends in Fig.\\,16 from the right to the left indicate a decrease of the radii for the distances d\\,=\\,6\\,kpc and 1\\,kpc, by about 60\\,R$_{\\rm \\odot}$ (15\\%) and\n 11\\,R$_{\\rm \\odot}$ (15\\%) by an LTV visual brightness decrease of 0\\fm11 in $V$, and an LTV blueing of 0\\fm15 in $(B-V)$, respectively.\n\nWe derived the contraction rate during the BL evolution of HD\\,179821 (roughly between JD\\,24\\,49000 and JD\\,24\\,53200) for a distance of 6\\,kpc: the result is $\\sim$\\,2.0\\,R$_{\\rm \\odot}$\\,y$^{-1}$.\n\n\\section{Theoretical and speculative explanations for two observational highlights.}\n\nBelow we discuss two speculative subjects. Firstly, the one raised in Sect.\\,2.3.\nWe  believe that all convective stars, such as YHGs, are subject to a random process that may be identified as the convection dynamics acting on the periodic pressure waves by the $\\kappa$-mechanism of the He-ionization zone (Fadeyev 2011). This principle has been used by Icke et al. (1992) for their numerical study of the instability of post-AGB stars, producing model light curves with some similar characteristics to those of YHGs.\n\nIt should be noted that Wisse (1979) and Icke et al. (1992) did not include the LTVs (representing the BL and RL evolutions) in their calculations, on which the much shorter quasi-periods are superimposed. This means that the calculations should have been done with one more variable. Based on a comparison of the main characteristics of the observed time series of the four YHGs, and those of the calculated time series without LTVs, we conclude that LTVs have obviously little effect: the observed pulsations also show a weakly chaotic character just like the models.\n\nNow we discuss observations detailed in Sects.\\,2.4.--2.6. and in Appendix\\,D:\\,2.6., namely indications for the presence of a variable high-opacity layer in the atmospheres of YHGs, responsible for a wavelength-dependent absorption.\nDeviating photometric and calculated stellar properties emerged from analyses of individual light and colour curves, for example by means of the $V$\/$(B-V)$ and Ampl\\,$V$\/Ampl\\,$(B-V)$ diagrams. As spectral temperatures are independent of continuum absorption, they always appear to be higher than photometric ones. The differences decrease from about 3000\\,K to a few 100\\,K from high (about 8000\\,K) to low (about 4500\\,K) temperatures, respectively.\n\nAs a result of the abnormally high amplitude ratios Ampl\\,$V$\/Ampl\\,$(B-V)$, the calculated temperatures for HR\\,8752, for example, are about 200\\,K too low (compared to the results of Nieuwenhuijzen et al. 2012: their Fig.\\,2 based on MK spectral types and LTE effective temperatures), and the calculated stellar radii in the maximum are largest: 946\\,R\/R$_{\\rm \\odot}$ and in the minimum smallest: 866\\,R\/R$_{\\rm \\odot}$, respectively. The dimensions on the other hand should be the other way around (estimated relative and absolute errors are about 100\\,R\/R$_{\\rm \\odot}$, compared to the results of Nieuwenhuijzen et al. (2012) in their Fig.\\,20).\nThe photometric observations on which our calculations are based, are from Walker (1983); Ferland priv. comm. to Lambert \\& Luck (1978); Moffett \\& Barnes (1979).\n\nA discussion on variable transparency sources in stellar atmospheres is given by Lobel (1997, Ch.1.).\nFor example, it is known that the opacity depends on the density of partially ionized hydrogen gas. The quasi-periodic pulsations of YHGs expand and compress their very extended atmospheres causing variable opacity and absorption of radiation that yield temporal changes of the stellar T$_{\\rm eff}$ and mass-loss rate. During the pulsation-compressions the increased surface gravity raises the gas density of extended layers in the optically thin portions of the outer atmosphere (i.e. $\\tau_{\\rm R}$<$2\/3$ where the local radiation temperature T$_{\\rm rad}$ is not in equilibrium with T$_{\\rm gas}$), consequently altering the selective absorption of incoming radiation and producing significant optical depth effects.\nLobel (2001) discussed the possibility that the T$_{\\rm eff}$ variations with pulsations can alter the temperature of the incoming radiation field (with T$_{\\rm rad}$ variations proportional to the T$_{\\rm eff}$ changes) inside these atmospheric layers, further shifting the local (LTE) thermal H ionization balance due to (non-LTE) photo-ionization. The deviations from an average T$_{\\rm rad}$ occur because of variations in the local opacity sources at different atmospheric heights. The modified partial H ionization alters the layer opacity. But it also changes the overall compressibility (or bulk modulus) through $\\Gamma_{1}$ and hence its local dynamic stability in the stellar atmosphere that may cause the unreliability of $(B-V)$ to derive an accurate temperature.\n\n\n \\section{Conclusions.}\n\nThe most important novelties to come from the present study can be summarised as follows.\n\n1. Spectroscopic temperatures (T(Sp)) are always higher than photometric temperatures (T(phot)) independent of the temperature scale used. Differences (a few thousand K near 8000\\,K) decrease towards lower temperatures (a few hundred K near 4200\\,K), and must be due to the presence of a temperature-dependent high-opacity layer in the atmosphere. After all, T(Sp) is relatively insensitive to continuum absorption, contrary to T(Phot). As a result, $B$ and $V$ magnitudes are inconsistent with respect to each other; $B$ has suffered from more absorption than $V$.\nObviously, the higher the temperature, the higher the opacity of that atmospheric layer. It goes without saying that this was caused by increased mass loss and subsequently by a higher gas density of that layer rendering a too low T(Phot) (Sects.\\,2.4.--2.6.).\n\n2. The analysis of the LTVs of $\\rho$\\,Cas, HD\\,179821, and HR\\,5171A, and confirming the LTV of HR\\,8752 by Nieuwenhuijzen et al. 2012, revealed that they mainly consist of RL and BL evolutions. These evolutions are responsible for the alternating T$_{\\rm eff}$ variations of 1000\\,K--4000\\,K. The corresponding observed $(B-V)$ ranges are 0\\fm2--0\\fm6 (Sects.\\,3.2. and 3.4.1.).\n\n3. The above suggests that HD\\,179821 is likely a YHG, not a RSG, nor a post-AGB star. If correct, the star should be located at a large distance (probably larger than 5\\,kpc), and far above the galactic plane, creating a kinematical problem (Sect.\\,3.3.) as already mentioned by Arkhipova et al. (2009).\n\n4. In light of the results described in (2) above, a schematic sequence of RL and BL evolutions could be defined for HR\\,5171A in Fig.\\,14, separated by the 1975 and 2000 eruptions. The most plausible explanation for the persistent reddening episode in 1960--1981 of HR\\,5171A is that it consists of an RL evolution between 1960 and the 1975 eruption. Thereafter, a severe absorption episode should have happened right after this eruption by a massive shell ejection, simultaneously with a BL evolution (Sect.\\,3.4.1.).\n\n5.  Quasi-periods, pulsation amplitudes, and stellar radii decrease and increase during BL and RL evolutions, respectively (Sects.\\,2.5., 3.1., 3.3, 3.4.3., 3.5., Figs.\\,11 and 13 and Appendix\\,K:3.3.).\n\n6. Based on light-curve markers, a new but weak eruption of $\\rho$\\,Cas was identified which happened in 2013, and which was independently spectroscopically confirmed by Aret et al. (2016; Sect. 2.2. in the present paper).\n\n7. The light variations of HR\\,5171A of the order of a year are due to pulsations alone. Any significant contribution to the light variation of a distorted primary by a possible much smaller contact companion is out of the question (Sect.\\,2.3.; Chesneau et al. 2014, their Fig.\\,8). However, we cannot exclude an unobserved companion.\n\n8. The photometric observations reveal that each pulsation is unique, any predictability on the morphology and duration of the next pulsation is impossible. These facts are in accordance with the light-curve models of Icke et al. (1992) and Wisse (1979) for evolved long-period variables (Sect.\\,2.3.). Based on the model light curves of  Icke et al. (1992), we characterize the pulsations of the four YHGs as `weakly chaotic' rather than `chaotic'. For both types of model light curves, the incessant sequences of photometric pulsations appear coherent, which agrees with the observations (Figs.\\,2 and 3).\n\n9. The membership of HR\\,5171A to the stellar cluster Gum48d at a distance of 3.6\\,kpc is doubtful. Its distance is probably of the order of 1.5\\,kpc\\,$\\pm$\\,0.5\\,kpc (Sect.\\,2.9.). If this supposition is correct, it would prove that HR\\,5171A is not amongst the biggest stars known. For $\\rho$\\,Cas, a shorter distance is preferred as well: 2.5\\,kpc\\,$\\pm$\\,0.5\\,kpc, instead of the accepted one of 3.1\\,kpc\\,$\\pm$\\,0.5\\,kpc, (Sect.\\,2.9.).\n\n10. We would like to finish this paper with the motto of the Royal Astronomical Society, London, 1820:\n`Quicquid nitet notandum', or `Whatever shines should be observed'.\nSee Appendix\\,L: Message to observers.\n\n\\begin{acknowledgements}\nA.L. acknowledges support in part by the Belgian Federal Science Policy Office under contract No.\\,BR\/143\/A2\/BRASS and funding received from the European Union's Framework Programme for Research and Innovation Horizon 2020 (2014--2020) under the Marie Sklodowska-Curie grant Agreement No.\\,823734.\nG.W.H. thanks Lou Boyd for his support of Fairborn Observatory and acknowledges long-term support from Tennessee State University and the State of Tennessee through its Centers of Excellence Program.\nWe acknowledge with thanks all the variable star observers worldwide from the AAVSO, especially S. Otero, VSX Team, for some invaluable comments on this paper and offering his historical data of HR\\,5171A which were used in this paper, and which will be of great benefit for future research.\nWe like to thank Mr. Roland Timmerman for making Fig.\\,1.\nAn anonymous referee is gratefully acknowledged for the invaluable comments and suggestions, which improved the presentation of this paper.\n\nWe would like to thank the following colleagues for their stimulating and fruitful correspondence since 2013, on various subjects addressed in this paper: Prof.dr. R.M. Humphreys, Dr. R.D. Oudmaijer, Dr. F. van Leeuwen (Hipparcos), Dr. B. Mason (Washington Double Star Cat.), Dr. J. Grindlay (Harvard DASCH scanner), Prof. dr. V. Icke, Dr. J. Lub, Prof.dr. J.W. Pel, Dr. D. Terrell, Prof.dr. A. Witt (Blue Luminescence of PAH molecules), Mr. E. Budding and Dr. N. I. Morrell (spectrograms of HR\\,5171A), Dr. C. Sterken, Dr. A. Meilland and Dr. E. Chapellier. This research made use of the excellent near-infrared (JHKL) data obtained at the SAAO, and discussed by Chesneau et al. (2014).\n\n  \\end{acknowledgements}\n\n\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}