diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdkqh" "b/data_all_eng_slimpj/shuffled/split2/finalzzdkqh" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdkqh" @@ -0,0 +1,5 @@ +{"text":"\\section*{Introduction}\n\nTechnological innovation has lead to a world full of data of an increasingly\ngrowing dimension. These data in turn contain information, the extraction\nof which is a basic task of Statistics (c.f. \\citet{lindsay2004}).\nAn important type of information is the kind of interdependence among\nvariables being represented by data. This calls for statistical means\nof extracting, quantifying and, if possible, modeling such interdependence.\nAt the very least, coefficients that somehow summarize the type and\nintensity of multivariate interdependence are very desirable in applied\nscience.\n\nThe introduction of the correlation concept by Francis Galton (1822-1911)\nhad a tremendous impact on many sciences due to its straightforward\ninterpretation as a measure of ``partial causation'' or ``average\nassociation'', as summarized in a single parameter. However elementary\nthis concept now may seem, it was welcomed as an important scientific\ncontribution at the end of the XIX century. As \\citet{pearson_life_2011}\nwrites, it did ``open to quantitative analysis wide fields of medical,\npsychological and social research {[}...{]}, {[}it{]} was to replace\nnot only in the minds of many of us the old category of causation,\nbut deeply to influence our outlook on the universe''. \n\nKarl Pearson would develop the original correlation coefficient of\nGalton into the widely used product-moment correlation coefficient.\nGiven the first paradigmatic step, new implementations of the concept\nwould appear, in the form of other association coefficients more adequate\nfor specific applications in psychology and the social sciences: Spearman's\n$\\rho$, Kendall's $\\tau$, Ginni's $\\gamma$, Blonqvist's $q$, etc.\n(The reader is referred to \\citet{joe_relative_1989} for more coefficients).\nThese are coefficients intended to represent the degree of association\nbetween two random variables.\n\nLater on, \\citet{renyi_measures_1959} would attempt to give some\nmathematical rigor to the concept of dependence, providing ``seven\nrather natural postulates which should be fulfilled by a suitable\nmeasure of dependence''. R\\'enyi's work would be revised by \\citet{schweizer_nonparametric_1981},\nwho made some ``reasonable modifications'' to the postulates, since\nthey were found to be too restrictive. Additionally, \\citet{schweizer_nonparametric_1981}\nused the concept of copulas to introduce a number of measures of pair-wise\ndependence which fulfilled their new postulates. With these conceptual\ntools (i.e. a set of reasonable postulates and the unifying concept\nof copulas), \\citet{wolff1980n} extends the measures of dependence\nbetween two variables given by \\citet{schweizer_nonparametric_1981},\nand proposes an extension to more than two variables of Spearman's\n$\\text{\\ensuremath{\\rho}}$. This course of action has been further\nfollowed and developed by \\citet{jaworski_copula-based_2010}. Indeed,\n\\citet{jaworski_copula-based_2010} introduce a series of measures\nthat can be considered as extension to more that two variables of\nsome of the well-established, pair-wise measures of dependence mentioned\nabove.\n\nAnother course of action, traceable back to \\citet{linfoot1957informational},\nis to use entropy or mutual information as association coefficient.\n\\citet{joe_relative_1989} proposes a number of measures of this type\nthat apply to more than two variables; \\citet{pena2007dimensionless}\nintroduce a measure which adjusts itself to dimension, so as to compare\nthe intensity of association of two vectors of different dimensions.\n\\citet{Micheas2006765} deal with the general case of $\\varphi$-dependence,\nof which mutual information is one particular case. The intensity\nof association is measured by \\citet{Micheas2006765} in terms of\nthe deviance of the joint distribution from the distribution given\nby the product of the marginal distributions (the independence case).\nThe specific definition of deviance depends on the specific selection\nof function $\\varphi:\\left[0,+\\infty\\right)\\rightarrow\\mathbb{R}$,\nwhich is continuous and convex, satisfying some basic conditions. \n\nApart from their theoretical interest, measures of dependence for\nmore than two variables are required and sought in applied research.\nIn the area of \\emph{neuronal science}, an influential theory of behavior\nintroduced by \\citet{hebb_organisation_1949} suggests that ``fundamental\ninsight into the nature of neuronal computation requires the understanding\nof the cooperative dynamics of populations of neurons'' (\\citet{grun_analysis_2010},\nchapter 12), and further evidence in the course of the years has lead\nbrain theorists to build models that ``rely on groups of neurons,\nrather than single nerve cells, as the functional\\emph{ }building\nblocks for representation and processing of information'' (\\citet{grun_analysis_2010},\npreface); this has lead to the development of techniques to quantify\nbeyond pair-wise association in that research area. Concerning applied\n\\emph{atmospheric research}, \\citet{bardossy_copula_2009} in the\ncontext of daily precipitation modeling, and \\citet{bardossy_multiscale_2012}\nin the context of downscaling, have found evidence that explicit quantification\nof interactions among more than two variables, and their proper incorporation\ninto modeling and forecasting, may be of an importance hitherto unexplored:\npredictions based on statistical models can be otherwise severely\nbiased, particularly for very high (or extreme) values of the multivariate\nprocess modeled. More recently, \\citet{ellipticalSpatialRodriguezBardossy}\ninvestigate the consequences for inference of ignoring multivariate\ninterdependence in the context of Spatial Statistics, and propose\na model that can deal with this type of interdependence explicitly.\nThe present paper comprises the theoretical basis for the work of\n\\citet{ellipticalSpatialRodriguezBardossy}. In the area of \\emph{finance},\n``herding'' behavior, the degree to which several economic actors\nbehave as a herd (\\citet{Dhaene2012357}), doing basically the same\nthing, is important for estimation of loss risks: If a single underlying\nfactor or small number of factors are inducing a high degree of herd\nbehavior, financial assets practically independent or very loosely\ncorrelated with each other can interact \\emph{en bloc}, rendering\nportfolio diversification ineffective. As \\citet{Dhaene2012357} indicate,\npair-wise correlations, or a measure based on these, may be misleading\nin this case. \\citet{RePEc:ner:leuven:urn:hdl:123456789\/410070} present\na related interdependence measure for aggregating risks. \n\nIn a recent paper, \\citet{reimherr2013} note that most of the theory\non measures of association has left out the important issue of interpretability\nof the measures for the research at hand. These authors argue (correctly,\nin our opinion) that the lack of interpretability limits the their\nuse as summary tools. This problem is greatly exacerbated if what\nwe intend to quantify or represent is the association among several\nvariables. In a more general manner, the interpretability of parameters\nand coefficients of a statistical model has been considered an important\ncharacteristic of the model by \\citet{cox1995relation}. \n\nThis paper deals with some of the issues inherent in formulating interaction\ncoefficients that pertain to more than two variables. We propose an\napproach for dealing with these issues. Section \\ref{sec:Difficulties-of-defining}\nintroduces some issues that one encounters when dealing with measures\nof interaction for more than two variables. Section \\ref{sec:Interaction-parameters-versus-manifestations}\nstates the approach we suggest for dealing with these issues: to discriminate\nbetween interaction ``parameters'' and interaction ``manifestations''.\nWe illustrate what we mean by the names \\emph{interaction parameter}\nand \\emph{interaction manifestation}. Section \\ref{sec:The-Lancaster-Interaction}\nintroduces joint cumulants and Lancaster Interactions. The relation\nbetween the two is exhibited, and a justification of joint cumulants\nas legitimate extensions to covariance coefficients indicated. Section\n\\ref{sec:Interaction-manifestations-in-terms} exhibits the relation\nbetween joint cumulants and some illustrative interaction manifestations,\nas defined in this paper. Section \\ref{sec:Illustration:-Runoff-to}\nillustrates the ideas presented, in that a specific model is introduced,\nand the ideas of this paper applied to simulated data. In section\n\\ref{sec:Discussion}, a discussion of the results is provided. \n\n\n\\section{\\label{sec:Difficulties-of-defining}Difficulties of defining a measure\nof multivariate interaction}\n\n\n\\subsection{\\label{sub:Interpretability}Interpretability }\n\nFor a two dimensional dataset, interpretability of a dependence coefficient\nis aided by the possibility of plotting the data. One looks at several\ndatasets and computes the respective coefficient of dependence. After\nmany such data sets, one has an idea of what, say, a correlation coefficient\nwith a value of $-0.8$ stands for. This visual aid is still possible\nfor three dimensional datasets, but is not available for higher dimensions.\nAssuming we have a coefficient of interdependence, $\\lambda$, applicable\nto multivariate vectors; how is one supposed to interpret a value\nof $\\lambda\\left(X_{1},X_{2},X_{3},X_{4}\\right)=-0.8$? Can one visualize\na dataset producing such a coefficient, so as to relate it to the\nphenomenon one is investigating? \n\nIt has been claimed that major advances in the science of statistics\nusually occur as a result of the theory-practice interaction (\\citet{box1976}),\nand that the parameters of a model should have clear subject-matter\ninterpretations (\\citet{cox1995relation}). These statements suggest\nthat interaction parameters as mere abstract constructions will not\nfind much application in statistical modeling, unless one can ``paraphrase''\ntheir meaning and relate them to the problem at hand. \n\nOur approach to interaction quantification and modeling consists in\ndiscriminating between interaction \\emph{manifestations} and interaction\n\\emph{parameters}. So, we can focus on quantification and modeling\nof what \\emph{really interests us} about dependence in data (i.e.\nits subject-matter relevant manifestation), while trying to reproduce\nsuch manifestations with as few parameters as possible. \n\n\n\\subsubsection*{Relation to the ``probability inversion'' technique in Probabilistic\nRisk Assessment}\n\nAn analogous approach has found successful application in the area\nof probabilistic risk assessment (\\citet{bedford2001probabilistic}).\nWith the aid of mathematical models, it is often possible to predict\n(approximately) what the consequences of a given event may be. This\nmathematical model has parameters that ideally should be calibrated\non the basis of past data. However, absence of data for certain events\n(e.g. a nuclear accident in a given region) makes the use of expert\nknowledge necessary, whereby the model parameters are to be estimated\non the basis of the experience of a group of experts. Experts usually\ncannot give an adequate direct evaluation of the joint probability\ndistribution of the model parameters, or \\emph{target variables}.\nHence each expert is asked to express his uncertainty judgments in\nterms of \\emph{elicitation variables}, i.e. \\emph{observable quantities\n}within the area of his\/her expertise. A target variable-set for the\nmodel is then recovered, such that the elicitation variables produced\nby the mathematical model look as similar as possible like the elicited\nvariables provided by the experts. This is an inverse problem, labeled\n``probabilistic inversion''. The interested reader is referred for\nmore details to (\\citet{bedford2001probabilistic,Du20061164}) and\nthe references therein.\n\nWe suggest in section \\ref{sec:Interaction-parameters-versus-manifestations}\na course of action that is analogous to ``probabilistic inversion''\nfor the problem of interactions quantification and modeling. \n\n\n\\subsection{\\label{sub:High-parametric-dimensionality}High parametric dimensionality}\n\nA second issue when defining an interaction coefficient, is the issue\nof high dimensionality. As dimension of the random vector under analysis\nincreases, a naive use of interaction coefficients becomes prohibiting.\nFor example, the correlation matrix of a 10-dimensional random vector\nis an array having $45$ correlation coefficients. Assume symmetry\non the variables with respect to the association coefficient (i.e.\nthe order of the variables plays no role on the coefficient's value):\nIf, for the same 10-dimensional vector, one intends to consider 3-wise,\n4-wise and 5-wise \\textquotedbl{}correlation coefficients\\textquotedbl{},\nthe corresponding arrays would have 450, 4500, and 45000 coefficients,\nrespectively. \n\nHence, it is necessary to be able to select judiciously the interaction\nparameters with which to work, and impose reasonable constraints on\nthem.\n\nAnother aspect that can be considered a sort of ``curse'' of dimensionality,\nis the coefficient of interdependence to use: there are too many features\nthat multivariate datasets can exhibit. \n\nIn the one-dimensional case, parameters such as mean, standard deviation,\nskewness and kurtosis (basically, the first four cumulants) give a\nlot of information about the distribution of data, provided these\ndata come from an unimodal distribution. Those parameters (mean, skewness\ncoefficient, etc.) describe data to some extent, since they can be\nreadily connected to specific questions about data: the location of\ndata, how informative this location about data is, how symmetric the\ndistribution is, to what extend can one expect values very far away\nfrom the mean. As a reference one may have in mind these characteristics\nfor the normal distribution. \n\nBut as dimension grows, one must focus on that feature of data interaction\nwhich is most connected with the research questions at hand, rather\nthan on an abstract dependence coefficient.\n\n\n\\subsubsection{How high dimensionality is dealt with in the realm of Spatial Statistics}\n\nWe present now an example of how the issue of high dimensionality\nhas been addressed in the context of Spatial Statistics. This will\ngive us a basis method from which to generalize. \n\nIn the area of spatial statistics (see, for example \\citet{cressie_statistics_1991,cressie2011statistics,diggle2007model}),\nthe studied random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$ spans hundreds\nor thousands of components, each of which component represents the\nvalue of an environmental process at a given location $j=1,\\ldots,J$.\nThe way high dimensionality is addressed in spatial statistics is\nan apt introduction for the method we advocate in this paper. We give\nhere a very basic form of a spatial statistical model, since it suffices\nfor our introducing purposes.\n\nOne focuses on the correlation between every two components of $\\mathbf{X}$.\nThe covariance among every two components, $\\left(X_{i},X_{j}\\right)$,\nof $\\mathbf{X}$, is expressed as a function $Cov\\left(d\\right)$\nof the distance between the locations represented by these two components,\n$d\\geq0$. The covariance function $Cov\\left(d\\right)$ must be such\nthat the resulting covariance matrix is positive definite. To this\nend there are a number of covariance functions often used in practice,\nfor example, one popular covariance function is the powered exponential\none, \n\\begin{equation}\nCov\\left(d\\right)=\\sigma_{0}^{2}.I\\left(d=0\\right)+\\sigma_{1}^{2}\\exp\\left(-\\left(d\/\\theta_{1}\\right)^{\\theta_{2}}\\right)\\label{eq:Power-exp_cov}\n\\end{equation}\nwhere $\\theta_{1}>0$, $0<\\theta_{2}\\leq2$, $\\sigma_{0}^{2}\\geq0$,\n$\\sigma_{1}^{2}\\geq0$ are the covariance function parameters.\n\nNote that:\n\\begin{enumerate}\n\\item Function (\\ref{eq:Power-exp_cov}) allows to have the covariance between\nevery two components of $\\mathbf{X}$ as a function of the distance\nbetween the locations these components represent, and only 4 additional\nparameters. In this way, the whole dependence structure of $\\mathbf{X}\\in\\mathbb{R}^{J}$\n(with $J>>2$) is \\emph{low dimensionally} obtained, built on the\nbasis of 2-dimensional dependence coefficients. \n\\item The interesting \\emph{dependence manifestation} to recover is covariance\nbetween every two components of $\\mathbf{X}$, whereas the (interaction)\nparameters to estimate are the function parameters, $\\theta_{1},\\theta_{2},\\sigma_{0}^{2}$\nand $\\sigma_{1}^{2}$. This is entirely analogous to the probability\ninversion technique mentioned in section \\ref{sub:Interpretability}:\ncovariance takes the place of the elicitation variables, whereas the\ncovariance function parameters are the target variables.\n\\item There is a functional relation between $\\theta_{1},\\theta_{2},\\sigma_{0}^{2},\\sigma_{1}^{2}$\nand the dependence manifestation. Covariance can be written in terms\nof the (interaction) parameters, $\\theta_{1},\\theta_{2},\\sigma_{0}^{2}$\nand $\\sigma_{1}^{2}$.\n\\end{enumerate}\nItems 1 through 3 summarize a technique to tackle the problem of high\ndimensionality in an ingenious low-dimensional way. The issue of interpretability\ngoes relatively unnoticed, since in this case parameters have a relatively\nstraightforward interpretation: $\\sigma_{0}^{2}$ represents a micro\nscale variability of the environmental process; $\\sigma_{0}^{2}+\\sigma_{1}^{2}$\nrepresents the variance of the marginal distribution of each component\n$X_{j}$ of $\\mathbf{X}$; $\\theta_{1}$ (often called ``range'')\nrepresent the distance at which correlation between data from two\nlocations is relatively insignificant. The parameter $\\theta_{2}$\nmight even receive a suitable interpretation, depending on the context.\n\nIn the next section, this approach is extended to deal with the interdependence\namong more than two variables at a time, keeping basically the same\nideas.\n\n\n\\section{\\label{sec:Interaction-parameters-versus-manifestations}Interaction\nparameters versus interaction manifestations}\n\nThe approach we advocate in this paper can be summarized as follows:\nfirst select an interaction ``manifestation'' relevant for the research\nin question. Then fit (low-dimensional) interactions ``parameters''\nthat make the fitted distribution reproduce, as close as possible,\nthe observed interaction manifestation. In this way, we circumvent\nthe issues of interpretability and high dimensionality mentioned above. \n\nBy interaction manifestation, we mean any function of more than one\ncomponent of the random vector analyzed, $\\mathbf{X}\\in\\mathbb{R}^{J}$,\nwhich can be interpreted as relevant for the research objectives at\nhand. For the sake of illustration:\n\\begin{enumerate}\n\\item The distribution of the sum of subsets of components of a random vector.\nIn the context of financial analysis, this sum is readily interpreted\nas ``risk'' (see also section \\ref{sec:Illustration:-Runoff-to}\nbelow). \n\\item The joint distribution of subsets of components, or the probability\nof trespassing simultaneously a threshold defined for each component.\nThis is useful in many applications. For example, in the context of\nseries systems reliability, such trespassing probability is the probability\nof ``failure''. \n\\item Differential entropy, any information-based dependence measure, or\nany of the copula-based generalizations to correlation measures studied\nby \\citet{jaworski_copula-based_2010}, of subsets of components.\nDepending on the specific research carried out, these may have subject-matter\ninterpretations, or can readily provide the versed researcher of a\nspecific area with a summary picture of the dependence in the data. \n\\end{enumerate}\nInteraction manifestations are interesting for the problem at hand,\nwe would like our model to reproduce them properly. But they are not\nvery helpful for building a model that integrates them, let alone\na low-dimensional model. \\emph{If we had }interaction parameters or\ncoefficients which:\n\\begin{enumerate}\n\\item Provide us with an idea of the number of variables interacting within\nthe random vector analyzed, $\\mathbf{X}\\in\\mathbb{R}^{J}$.\n\\item Can be somehow (functionally) connected with the interaction manifestations\nthat are interesting for the research carried out. \n\\item Can be built into a parametric or semi-parametric model. This would\nimmediately open up the possibility of a low-dimensional model, via\na judicious selection of assumptions and\/or constraints on the interaction\nparameters.\n\\end{enumerate}\n\\emph{Then we could} proceed, in the manner of an inverse problem,\nas follows:\n\\begin{enumerate}\n\\item We find data-based estimates or approximations to the interesting\ninteraction manifestations\n\\item We fit the interactions parameters so as to match best the observed\ninteraction manifestations\n\\end{enumerate}\nIn the next section, we introduce a reasonable interaction measure,\nand through it, a reasonable type of interaction parameter with which\none can work along the lines above; namely the joint cumulant. We\nclaim that using joint cumulants as building blocks of a multivariate\nstatistical model allows for an adequate consideration of dependence,\nboth of pairs of variables, and of groups of more variables. \n\nIt might be argued that moments (and hence cumulants) of sufficiently\nhigh orders might not exist for the ``true'' probability distribution\nof the process under analysis. We would answer that such distributions\ncan always be sufficiently (i.e. for practical purposes) approximated\nby a distribution with existing moments of all orders. See, for example\n\\citet{1987}, where the authors introduce a semi-parametric model,\nsimilar to an Edgeworth expansion. This model possesses moments of\nall orders. Yet, under minimal conditions it can approximate \\emph{any}\ncontinuous distribution on $\\mathbb{R}^{J}$, provided sufficiently\nmany factors are added to the sum defining the model. Additionally,\n\\citet{del_brio_gramcharlier_2009,mauleon2000testing,perote_multivariate_2004}\npresent variants of the model of \\citet{1987}, and show how they\ncan be effectively applied to modeling heavy tailed data, both univariate\nand multivariate. \n\n\n\\section{\\label{sec:The-Lancaster-Interaction}The Lancaster Interaction Measure\nand Joint Cumulants}\n\nIn this section, the connection between the Lancaster Interaction\nmeasure of a random variable and its joint cumulants is established.\nTo our knowledge, this connection has not been pointed out before\nas a justification of joint cumulants as reasonable interdependence\nparameters.\n\n\n\\subsection{A review of Lancaster Interactions}\n\nWe review now the function called ``additive interaction measure''\nor ``Lancaster interaction measure'', introduced by \\citet{lancaster_chi-squared_1969}\nand later modified by \\citet{streitberg_lancaster_1990}. This function\ncan be built for every random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$,\nand has the property of being identically zero if any sub-vector of\n$\\mathbf{X}$ is independent of the other. \n\nAn additive interaction measure $\\Delta F$$\\left(\\mathbf{X}\\right)$\nis a signed measure determined by a given distribution $F\\left(\\mathbf{X}\\right)$\non $\\mathbb{R}^{J}$. Its defining characteristic is that it is equal\nto zero for all $\\mathbf{X}\\in\\mathbb{R}^{J}$, if $F\\left(\\mathbf{X}\\right)$\ncan be written as the non-trivial product of two or more of its (multivariate)\nmarginal distributions (\\citet{streitberg_lancaster_1990}). For example,\nif $J=4$ and $F$ can be written as $F_{124}F_{3}$, being $F_{124}$\nand $F_{3}$ the marginal distributions of $\\left(X_{1},X_{2},X_{4}\\right)$\nand $X_{3}$, respectively, then $\\Delta F\\left(\\mathbf{X}\\right)\\equiv0$,\nfor all $\\mathbf{X}\\in\\mathbb{R}^{J}$. \n\nAn alternative explanation is that $\\Delta F\\equiv0$, if one subset\nof $\\mathbf{X}$'s components is independent of another subset of\ncomponents. If $\\Delta F\\equiv0$, then $F$ is said to be \\textquotedbl{}decomposable\\textquotedbl{}. \n\nLancaster Interaction measure is defined by \n\\begin{equation}\n\\Delta F\\left(\\mathbf{X}\\right)=\\sum_{\\pi}\\left\\{ \\left(\\left(-1\\right)^{\\left|\\pi\\right|-1}\\left(|\\pi|-1\\right)!\\right)F_{\\pi}\\left(\\mathbf{X}\\right)\\right\\} \\label{eq:Lancaster_measure_definition}\n\\end{equation}\nwhere the sum is over all partitions, $\\pi$, of index set $C=\\left\\{ 1,\\ldots,J\\right\\} $. \n\nAn example will help clarify the notation: for index set $C=\\left\\{ 1,2,3,4\\right\\} $\nthere are 15 partitions, three of which are: $\\pi_{1}=\\left\\{ \\left\\{ 1\\right\\} ,\\left\\{ 2\\right\\} ,\\left\\{ 3,4\\right\\} \\right\\} $,\n$\\pi_{2}=\\left\\{ \\left\\{ 1,4\\right\\} ,\\left\\{ 2,3\\right\\} \\right\\} $,\n$\\pi_{3}=\\left\\{ \\left\\{ 1,2,3,4\\right\\} \\right\\} $. Their cardinalities\nare $\\left|\\pi_{1}\\right|=3$ , $\\left|\\pi_{2}\\right|=2$ and $\\left|\\pi_{3}\\right|=1$,\nrespectively. In general, a set of $J$ elements has a total of $B_{J}$\npossible partitions%\n\\footnote{The number $B_{J}$ is often called Bell's number.%\n}, where $B_{0}=B_{1}=1$ and any subsequent $B_{k>1}$ can be found\n(see e.g. \\citet{rota_number_1964}) by the recurrence relation $B_{k+1}=\\sum_{r=0}^{k}{k \\choose r}B_{r}$.\nThe reader is referred to the textbook of \\citet{AignerDiskrete}\nfor more on partitions and their enumeration.\n\nThe symbol $F_{\\pi_{1}}$ is further to be interpreted as \n\\begin{equation}\nF_{\\pi_{1}}\\left(\\mathbf{X}\\right)=F_{1}\\left(X_{1}\\right)F_{2}\\left(X_{2}\\right)F_{34}\\left(X_{3},X_{4}\\right)\\label{eq:FActorized_distribution_example}\n\\end{equation}\nthat is, the product of the (multivariate) marginal distributions\ndefined by partition $\\pi_{1}$. The same explanation holds at (\\ref{eq:Lancaster_measure_definition})\nfor any of the $B_{J}$ partitions, $\\pi$, of index set $C=\\left\\{ 1,\\ldots,J\\right\\} $. \n\nIt will be convenient to define partition operator $J_{\\pi}$, to\nbe applied to $F$ for a given partition $\\pi$, by \n\\begin{equation}\nJ_{\\pi}F\\rightarrow F_{\\pi}\\label{eq:partition_operator}\n\\end{equation}\nwhere $F_{\\pi}$ is as in the example at equation (\\ref{eq:FActorized_distribution_example}). \n\n\\citet{streitberg_lancaster_1990,streitberg_alternative_1999} shows\nan important result concerning $\\Delta F$: given a probability distribution\nfunction $F$, function $\\Delta F$ as in (\\ref{eq:Lancaster_measure_definition})\nis the \\emph{only} function built as a linear combination of products\nof (multivariate) marginal distributions of $F$, such that $\\Delta F\\left(\\mathbf{X}\\right):=0$,\nwhenever one subset of $\\mathbf{X}$'s components is independent of\nanother components subset. \n\nSince the interaction measure is defined in terms of a given distribution\n$F$, we can define the interaction operator:\n\\begin{equation}\n\\Delta=\\sum_{\\pi}\\left\\{ \\left(\\left(-1\\right)^{\\left|\\pi\\right|-1}\\left(\\left|\\pi\\right|-1\\right)!\\right)J_{\\pi}\\right\\} \\label{eq:interactions_operator}\n\\end{equation}\nwhich, upon application to the distribution in question, returns the\nadditive interaction measure. \n\n\n\\subsection{A review of Joint Cumulants}\n\nMoments and cumulants can be defined as constants summarizing important\ninformation about a probability distribution and sometimes, even determining\nit completely (cf. \\citet{kendall_advanced_1969}). In this section\nwe deal with random variables having a probability density function.\nThe development is also valid for discreet distributions, under simple\nmodifications. The reader is referred to \\citet{kendall_advanced_1969,muirhead_aspects_1982,billingsley_probability_1986,mccullagh_tensor_1987}\nfor more details on moments and cumulants. \n\nThe Cumulant Generating Function (c.g.f.), $K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)$,\nof a random vector, $\\mathbf{X}\\in\\mathbb{R}^{J}$, is defined as\nthe logarithm of the moment generating function (m.g.f.),\n\\begin{equation}\nK_{\\mathbf{X}}\\left(\\mathbf{t}\\right)=\\log\\left(M_{\\mathbf{X}}\\left(\\mathbf{t}\\right)\\right)=E\\left(\\exp\\left(\\sum_{j=1}^{J}t_{j}X_{j}\\right)\\right)\n\\end{equation}\nwhere $\\mathbf{t}\\in\\mathbb{R}^{J}$, assuming these functions exist.\n\nJoint cumulants are then defined to be the coefficients of the Taylor\nexpansion for $K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)$, \n\\begin{equation}\nK_{\\mathbf{X}}\\left(\\mathbf{t}\\right)\\sim\\sum_{r_{1=0}}^{\\infty}\\ldots\\sum_{r_{J}=0}^{\\infty}\\frac{\\kappa_{r_{1},\\ldots,r_{J}}.t_{1}^{r_{1}}\\ldots t_{J}^{r_{J}}}{r_{1}!\\ldots r_{J}!}\n\\end{equation}\nand hence can be found by differentiating $K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)$\nand evaluating at $\\mathbf{t}=\\mathbf{0}$,\n\\begin{equation}\n\\kappa_{r_{1},\\ldots,r_{J}}=\\frac{\\partial^{r_{1}+\\ldots+r_{J}}}{\\partial^{r_{J}}t_{J}\\ldots\\partial^{r_{1}}t_{1}}K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)\\mid_{\\mathbf{t}=\\mathbf{0}}\n\\end{equation}\nwhere $r_{j}\\geq0$ is a non-negative integer. An important particular\ncase is the covariance coefficient, or second order joint cumulant,\n\\[\n\\frac{\\partial^{2}}{\\partial t_{i}\\partial t_{j}}K_{\\mathbf{X}}\\left(t_{i},t_{j}\\right)\\mid_{\\left(t_{i},t_{j}\\right)=\\left(0,0\\right)}=cov\\left(X_{i},X_{j}\\right)\n\\]\n\n\nThe c.g.f. of a sub-vector $\\mathbf{Y}=\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)$,\nwith indexes in an index set, $j_{i}\\in I$, can be readily found\nin terms of that of $\\mathbf{X}$, by setting the indexes not corresponding\nto $\\mathbf{Y}$ to zero: \n\\begin{multline*}\nK_{\\mathbf{Y}}\\left(\\mathbf{s}\\right)=\\left(E\\left(\\exp\\left(\\sum_{i=1}^{k}s_{i}X_{j_{i}}\\right)\\right)\\right)=\\log\\left(E\\left(\\exp\\left(\\sum_{j=1}^{J}g_{j}\\left(\\mathbf{s}\\right)X_{j}\\right)\\right)\\right)=K_{\\mathbf{X}}\\left(g\\left(\\mathbf{s}\\right)\\right)\n\\end{multline*}\nwhere $g:\\mathbb{R}^{k}\\rightarrow\\mathbb{R}^{J}$, and \n\\[\ng_{j}\\left(\\mathbf{s}\\right)=\\begin{cases}\n1, & j\\in I\\\\\n0, & j\\notin I\n\\end{cases}\n\\]\n\n\nAn alternative definition for joint cumulants uses product moments\nas departing point (see, for example, \\citet{brillinger_time_1974}).\nLet $\\mathbf{X}\\in\\mathbb{R}^{J}$ be a random vector. For a set $\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right)$\nof $\\mathbf{X}$\\textasciiacute{}s components, where some sub-indexes\n$j_{r}$ may be repeated, consider joint moments \n\\[\nE\\left(X_{j_{1}}\\ldots X_{j_{d}}\\right)\n\\]\n\n\nConsider partition operator $J_{\\pi}^{*}$, analogous to (\\ref{eq:partition_operator}),\nrelated to each partition $\\pi$ of $\\left(j_{1},\\ldots,j_{d}\\right)$.\nThis operator converts $E\\left(X_{j_{1}}\\ldots X_{j_{d}}\\right)$\ninto the product of the factors determined by partition $\\pi$. \n\nFor example, for $d=4$ , $\\left(j_{1},j_{2},j_{3},j_{3}\\right)$\nand $\\pi=\\left\\{ \\left\\{ 1\\right\\} ,\\left\\{ 2,3\\right\\} ,\\left\\{ 4\\right\\} \\right\\} $,\none has partition components $v_{1}=\\left\\{ 1\\right\\} $, $v_{2}=\\left\\{ 2,3\\right\\} $\nand $v_{3}=\\left\\{ 4\\right\\} $. Upon application of $J_{\\pi}^{*}$,\nwe have, \n\\[\nJ_{\\pi}^{*}E\\left(X_{j_{1}}\\ldots X_{j_{4}}\\right)=E\\left(X_{j_{1}}\\right)E\\left(X_{j_{2}}X_{j_{3}}\\right)E\\left(X_{j_{3}}\\right)\n\\]\n\n\nIn the general case\n\n\\[\nJ_{\\pi}^{*}E\\left(X_{j_{1}}\\ldots X_{j_{d}}\\right)=\\prod_{v\\in\\pi}E\\left(\\prod_{j_{r}\\in v}X_{j_{r}}\\right)\n\\]\n\n\nThe alternative definition of joint cumulants can now be given.\n\nFor random variables $\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right)$, their\njoint cumulant of order \\emph{d} is given by, \n\\begin{multline}\ncum\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right):=\\sum_{\\pi}\\left\\{ \\left(\\left(-1\\right)^{\\left|\\pi\\right|-1}\\left(\\left|\\pi\\right|-1\\right)!\\right)J_{\\pi}^{*}\\right\\} E\\left(X_{j_{1}}\\ldots X_{j_{d}}\\right)\\label{eq:Joint_cumulants_Brillinger}\n\\end{multline}\n\n\nTwo examples are:\n\n\\begin{eqnarray*}\ncum\\left(X_{1},X_{2}\\right) & = & E\\left(X_{1}X_{2}\\right)-E\\left(X_{1}\\right)E\\left(X_{2}\\right)\n\\end{eqnarray*}\nand\n\\begin{multline*}\ncum\\left(X_{1},X_{2},X_{3}\\right)=E\\left(X_{1}X_{2}X_{3}\\right)-E\\left(X_{1}X_{2}\\right)E\\left(X_{3}\\right)-E\\left(X_{1}X_{3}\\right)E\\left(X_{2}\\right)\\\\\n-E\\left(X_{2}X_{3}\\right)E\\left(X_{1}\\right)+2E\\left(X_{1}\\right)E\\left(X_{2}\\right)E\\left(X_{3}\\right)\n\\end{multline*}\n\n\nHence joint cumulants can be seen, from a merely formalistic point\nof view, to form a kind of higher order covariance coefficient. The\nsecond order joint cumulant is just the typical covariance coefficient. \n\n\n\\subsection{Relationship between Lancaster Interactions and Joint Cumulants}\n\nThe similarity between (\\ref{eq:Lancaster_measure_definition}) and\n(\\ref{eq:Joint_cumulants_Brillinger}) is evident. Indeed, if we concentrate\nfor now on the case $\\mathbf{X}\\in\\mathbb{R}^{2}$, then \\citet{lehmann_concepts_1966}\nreports that: \n\\begin{multline}\nCov\\left(X_{1},X_{2}\\right)=cum\\left(X_{1},X_{2}\\right)=\\\\\n\\intop_{-\\infty}^{+\\infty}\\intop_{-\\infty}^{+\\infty}\\left[F_{12}\\left(x_{1},x_{2}\\right)-F_{1}\\left(x_{1}\\right)F_{2}\\left(x_{2}\\right)\\right]dx_{1}dx_{2}\\label{eq:Hoeffding_Formula}\n\\end{multline}\nunder the condition that $E\\left(\\left|X_{1}^{k_{1}}X_{2}^{k_{2}}\\right|\\right)<+\\infty$,\nfor $k_{j}=0,1$.\n\nThis equation is often called \\textquotedbl{}Hoeffding's formula\\textquotedbl{}\nsince it was first discovered by \\citet{hoeffding_masstabinvariante_1940}.\nOf course, the above equation can be written in terms of the Lancaster\ninteraction measure (\\ref{eq:Lancaster_measure_definition}), as\n\\begin{equation}\ncum\\left(X_{1},X_{2}\\right)=\\intop_{-\\infty}^{+\\infty}\\intop_{-\\infty}^{+\\infty}\\Delta F\\left(x_{1},x_{2}\\right)dx_{1}dx_{2}\\label{eq:Hoeffding_Formula2}\n\\end{equation}\n\n\nIt turns out that this equation can be extended to higher dimensions.\nLet $\\mathbf{X}\\in\\mathbb{R}^{J}$ be a random vector. As shown by\n\\citet{block_multivariate_1988}, we have that (page 1808):\n\\begin{equation}\ncum\\left(\\mathbf{X}\\right)=\\left(-1\\right)^{J}\\intop_{-\\infty}^{+\\infty}\\ldots\\intop_{-\\infty}^{+\\infty}\\sum_{\\pi}\\left\\{ \\left(\\left(-1\\right)^{\\left|\\pi\\right|-1}\\left(\\left|\\pi\\right|-1\\right)!\\right)F_{\\pi}\\right\\} d\\mathbf{X}\\label{eq:cumul_lancaster}\n\\end{equation}\nunder the condition that $E\\left(\\left|X_{j}^{J}\\right|\\right)<+\\infty$,\nfor $j=1,\\ldots,J$. Again, this is the same as saying that\n\n\\begin{equation}\ncum\\left(\\mathbf{X}\\right)=\\left(-1\\right)^{J}\\intop_{-\\infty}^{+\\infty}\\ldots\\intop_{-\\infty}^{+\\infty}\\Delta F\\left(\\mathbf{X}\\right)d\\mathbf{X}\\label{eq:cumul_lancaster-1}\n\\end{equation}\n\n\nThus, joint cumulants are equal (up to a known constant) to the integral\nof Lancaster Interaction measure; they are ``summary'' or ``integral''\nmeasures of additive interaction. To our knowledge, this connection\nhad not been pointed out elsewhere.\n\nIt goes without much explanation that the joint cumulants of a random\nvector $\\mathbf{X}$ vanish whenever a subset of the vector is independent\nof another, since then the integrating function is identically zero.\nThis property is well-known and oftentimes the reason why joint cumulants\nare used in practice (e.g. in \\citet{brillinger_time_1974,mendel_tutorial_1991}).\nThe inverse is true only if the distribution of $\\mathbf{X}$ is determined\nby its moments, which may or may not be a reasonable assumption, depending\non the application. Again, based on the work of \\citet{1987,perote_multivariate_2004,mauleon2000testing,del_brio_gramcharlier_2009},\nwe argue that this is not an extreme limitation to our approach, since\nall we are seeking is a good approximation to the distribution under\nanalysis. \n\nIn particular, whenever we have $cum\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right)\\neq0$,\nwhere no index $j_{k}$ is repeated, this means that one cannot decompose\nthe distribution of $\\left(X_{j_{1}},\\ldots,X_{j_{d}}\\right)$: At\nleast $d$ variables within $\\mathbf{X}$ are interacting simultaneously\nwith each other. \n\nOur theoretical contribution here is that joint cumulants are seen\nas the integral of the Lancaster interaction measure. As shown by\n\\citet{streitberg_lancaster_1990}, $\\Delta F$ is the only additive\nmeasure, built very elementarily with the marginal distributions of\nthe random vector, which vanishes whenever one subset of $\\mathbf{X}$'s\ncomponents is independent of another subset of components.\n\nWe have provided a theoretical basis for declaring joint cumulants\n``interaction parameters'', and the cumulant generating function\na ``dependence structure''. The functional character of the c.g.f.\nopens up the possibility of parametric modeling, with its respective\nlow-dimensionality advantage. It is just another way of defining a\nmodel, alternative to the density specification. \n\nWe shall see below, how the parameters of a model expressed as a c.g.f.\ncan be connected with some interesting interaction manifestations.\n\n\n\\section{\\label{sec:Interaction-manifestations-in-terms}Interaction manifestations\nin terms of interaction parameters}\n\nThe connection between interaction parameters (i.e. joint cumulants)\nand interaction manifestations relies on the concepts of the Edgeworth\nexpansion and the saddlepoint approximation to the density of a random\nvector. A brief review of these topics is provided at the appendix. \n\n\n\\subsection{Connection of dependence structure with interaction manifestations}\n\nWe shall show explicitly the connection of joint cumulants and the\nc.g.f. with three of the interaction manifestations listed at section\n\\ref{sec:Interaction-parameters-versus-manifestations}, which manifestations\nrefer to subsets of components, $\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)$,\n$1\\leq k\\leq J$, of the random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$.\nNamely: the distribution of the sum of components; parameters related\nto the joint probability of the components; and the differential entropy\nof the components.\n\nA relevant point here is that, except for the distribution of the\nsum of components, even with a lot of data at hand, estimation of\nthe interaction manifestations mentioned can be done only for (multivariate)\nmarginals of relatively low dimension, such as $k$ equal to 3, 4\nor 5. But armed with a sensible c.g.f., we can consistently integrate\nthese manifestations into the whole distribution (in much the same\nway as thousand of covariance coefficients are integrated into a Spatial\nStatistics model that spans thousands of variables). This we can attain\nwith the aid of the overarching dependence structure, that is, the\nc.g.f. \n\nAssume for the moment you have a reasonable type of c.g.f., that is,\none that seems reasonable for the problem at hand (for an illustration\nsee section \\ref{sec:Illustration:-Runoff-to}). \n\n\n\\subsubsection{\\label{sub:Connection-of-dependence-sumas}Connection of dependence\nstructure with Sums of components}\n\nGiven a random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$ representing\nthe variables under analysis, we are interested in the distribution\nof variable $S_{\\mathbf{X}}=\\sum_{i=1}^{k}X_{j_{i}}$, where $\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)$,\n$1\\leq k\\leq J$, is a sub-vector of the random vector $\\mathbf{X}\\in\\mathbb{R}^{J}$.\nThe distribution of $S_{\\mathbf{X}}$ is the interaction manifestation\nwe in which we are interested. We want to fit the distribution of\nthe whole vector, $\\mathbf{X}\\in\\mathbb{R}^{J}$, in such a way the\nwe fit this interaction manifestation properly.\n\n\\emph{One course of action} is to find the cumulants of $S_{\\mathbf{X}}$\nin terms of the joint cumulants of $\\mathbf{X}$, and then approximate\nthe density of $S_{\\mathbf{X}}$, by using the Edgeworth Expansion.\nSince $S_{\\mathbf{X}}$ is a one-dimensional random variable, one\ncan alternatively find research-relevant quantiles of its distribution\nby inverting the Edgeworth Expansion, i.e. by using the Cornish-Fisher\ninversion. \n\nTo find the cumulants of $S_{\\mathbf{X}}$, note that two of the properties\nof joint cumulants are \\citet{brillinger_time_1974}: symmetry and\nmulti-linearity. Symmetry means that $cum\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)=cum\\left(P\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)\\right)$\nfor any permutation $P\\left(j_{1},\\ldots,j_{k}\\right)$ of the indexes\n$\\left(j_{1},\\ldots,j_{k}\\right)$. Concerning multi-linearity, for\nany random variable $Z\\in\\mathbb{R}$, one has\n\\[\ncum\\left(Z+X_{j_{1}},\\ldots,X_{j_{k}}\\right)=cum\\left(Z,\\ldots,X_{j_{k}}\\right)+cum\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)\n\\]\n Combining these two properties, it can be shown that\n\\begin{equation}\n\\kappa_{r}\\left(S_{\\mathbf{X}}\\right)=cum\\left(\\underbrace{S_{\\mathbf{X}},\\ldots,S_{\\mathbf{X}}}_{r}\\right)=\\sum_{i_{1}=1}^{k}\\left[\\sum_{i_{2}=1}^{k}\\ldots\\left[\\sum_{i_{r}=1}^{k}cum\\left(X_{j_{i_{1}}},\\ldots,X_{j_{i_{r}}}\\right)\\right]\\right]\\label{eq:cumulants-of-sums}\n\\end{equation}\nwhere $\\kappa_{r}\\left(S_{\\mathbf{X}}\\right)$ denotes the \\emph{r}-th\ncumulant of random variable $S_{\\mathbf{X}}=\\sum_{i=1}^{k}X_{j_{i}}$.\nThen the interesting quantiles of $S_{\\mathbf{X}}$ can be (approximately)\nwritten in terms of the $\\kappa_{r}$ via the Cornish-Fisher inversion. \n\nAs the dimension $k$ of the sub-vector increases, this approach becomes\nimpractical, since the sum at (\\ref{eq:cumulants-of-sums}) comprises\ntoo many elements. Fortunately, knowing the c.g.f. of $\\mathbf{X}$\ntells much about the c.g.f. of sums of its components.\n\n\\emph{A second course of action} uses all the information provided\nby the c.g.f. and is now given.\n\nIn a somewhat more general context as before, consider a random vector\n$\\mathbf{X}=\\left(X_{1},\\ldots,X_{J}\\right)$. One wishes to study\nthe joint distribution of aggregated variables of the form:\n\\begin{eqnarray}\n\\xi_{1} & = & \\sum_{j_{1}\\in I_{1}}X_{j_{1}}\\nonumber \\\\\n\\xi_{2} & = & \\sum_{j_{2}\\in I_{2}}X_{j_{2}}\\\\\n\\vdots & \\vdots & \\vdots\\nonumber \\\\\n\\xi_{l} & = & \\sum_{j_{l}\\in I_{l}}X_{j_{l}}\n\\end{eqnarray}\n\n\nwhere $I_{k}$, for $k=1,\\ldots,l$ represent non-overlapping index\nsets such that\n\n\\[\nI_{1}\\cup\\ldots\\cup I_{l}=\\left\\{ 1,\\ldots,J\\right\\} \n\\]\n(Note that $S_{\\mathbf{X}}$ above is the specific case in which $I_{1}=\\left\\{ 1,\\ldots,J\\right\\} $).\n\nThe cumulant generating function of the $l$-dimensional vector so\nobtained is given by\n\\begin{multline}\nK_{\\mathbf{\\xi}}\\left(\\mathbf{t}\\right)=\\log\\left(E\\left(\\exp\\left(\\mathbf{t}.\\mathbf{\\xi}^{'}\\right)\\right)\\right)=\\\\\n\\log\\left(E\\left(\\exp\\left(t_{1}\\xi_{1}+\\ldots+t_{l}\\xi_{l}\\right)\\right)\\right)=\\\\\n\\log\\left(E\\left(\\exp\\left(t_{1}\\sum_{I_{1}}X_{j_{1}}+\\ldots+t_{l}\\sum_{I_{l}}X_{j_{l}}\\right)\\right)\\right)=\\\\\n\\log\\left(E\\left(\\exp\\left(g_{1}\\left(\\mathbf{t}\\right)X_{1}+\\ldots+g_{J}\\left(\\mathbf{t}\\right)X_{J}\\right)\\right)\\right)=\\\\\n\\log\\left(E\\left(\\exp\\left(g\\left(\\mathbf{t}\\right).\\mathbf{X}^{'}\\right)\\right)\\right)=K_{\\mathbf{X}}\\left(g\\left(\\mathbf{t}\\right)\\right)\\label{eq:cum_gen_fun_suma}\n\\end{multline}\n\n\nFunction $g:\\mathbb{R}^{l}\\rightarrow\\mathbb{R}^{J}$ is a vector\nfunction defined by\n\n\\begin{eqnarray}\ng\\left(\\mathbf{t}\\right) & = & \\left(g_{1}\\left(\\mathbf{t}\\right),\\ldots,g_{J}\\left(\\mathbf{t}\\right)\\right)\\nonumber \\\\\ng_{j}\\left(\\mathbf{t}\\right) & = & \\mathbf{t}.\\left(\\mathbf{1}\\left(j\\in I_{1}\\right),\\ldots,\\mathbf{1}\\left(j\\in I_{l}\\right)\\right)^{'}\\label{eq:transf_cums}\n\\end{eqnarray}\nwhere\n\\[\n\\mathbf{1}\\left(j\\in I_{k}\\right)=\\begin{cases}\n1, & j\\in I_{k}\\\\\n0, & j\\notin I_{k}\n\\end{cases}\n\\]\n\n\nIt is hence possible to find the cumulant generating function of random\nvector $\\mathbf{\\xi}\\in\\mathbb{R}^{l}$ in terms of that of the original\nvector $\\mathbf{X}\\in\\mathbb{R}^{J}$. If we know the c.g.f. of the\noriginal random vector $\\mathbf{X}$, then the cumulants, the cumulant\ngenerating function, and hence the approximate joint density of the\naggregated variables, via Saddlepoint approximation at (\\ref{eq:Saddlepoint})\nof $\\mathbf{\\xi}\\in\\mathbb{R}^{l}$ are also determined (see section\n\\ref{sec:Illustration:-Runoff-to}). We can use this fact in order\nto fit the modeol for $\\mathbf{X}$ in such a way that the interesting\ninteraction manifestation (the sums of components) are explicitly\nconsidered in the estimation. \n\n\n\\subsubsection{Joint probabilities of (multivariate) marginals}\n\nJoint marginal distributions are usually important interaction manifestations.\nGiven a sub-vector $\\mathbf{Y}:=\\left(X_{j_{1}},\\ldots,X_{j_{k}}\\right)$\nof $\\mathbf{X}$, in order to find probabilities of the form \n\\[\n\\Pr\\left(X_{j_{1}}\\geq x_{j_{1}},\\ldots,X_{j_{k}}\\geq x_{j_{k}}\\right)\n\\]\none should in principle integrate expression (\\ref{eq:Saddlepoint}),\nfor the c.f.g. of $\\mathbf{Y}$.\n\nIn the uni-variate case, it is a well-established practice \\citet{Huzurbazar_saddlepoint1999}\nto employ instead an accurate approximation to that integral, which\nis due to \\citet{lugannani_rice1980}. Namely, in the univariate case,\nwe have: \n\\begin{multline}\nF_{X}\\left(x_{0}\\right)\\approx\\intop_{-\\infty}^{x_{0}}\\frac{\\exp\\left(K_{X}\\left(\\hat{\\lambda}\\left(x\\right)\\right)-x\\hat{\\lambda}\\left(x\\right)\\right)}{\\left(2\\pi\\right)^{1\/2}\\left(\\frac{d^{2}K_{X}\\left(\\mathbf{\\lambda}\\right)}{d\\lambda^{2}}\\mid_{\\lambda=\\hat{\\lambda}\\left(x\\right)}\\right)^{1\/2}}dx\\\\\n\\approx\\Phi\\left(r\\right)+\\phi\\left(r\\right)\\left\\{ \\frac{1}{r}-\\frac{1}{q}\\right\\} \\label{eq:Lugganani}\n\\end{multline}\n\n\nWhere $\\hat{\\tau}$ is such that $K_{X}^{'}\\left(\\hat{\\tau}\\right)=x_{0}$,\nand: \n\\begin{eqnarray*}\nr & = & sign\\left(\\hat{\\tau}\\right)\\left\\{ 2\\left[\\hat{\\tau}x_{0}-K_{X}\\left(\\hat{\\tau}\\right)\\right]\\right\\} ^{\\frac{1}{2}}\\\\\nq & = & \\hat{\\tau}\\left\\{ \\frac{d^{2}K_{X}\\left(\\lambda\\right)}{d\\lambda^{2}}\\mid_{\\lambda=\\hat{\\tau}}\\right\\} ^{\\frac{1}{2}}\n\\end{eqnarray*}\n\n\nThus, one must not perform the numerical integration at all. \n\nFor the multivariate case, \\citet{kolassa2010multivariate} have provided\na generalization of the Lugannani-Rice formula, which produces an\napproximation to probability $\\Pr\\left(\\mathbf{Y}\\geq\\mathbf{y}\\right)$\nof order $O\\left(n^{-1}\\right)$, for $\\mathbf{X}\\in\\mathbb{R}^{J}$.\nThis formula is extremely complicated and writing it here will most\nlikely obscure rather than clarify anything. Only the probability\ndistribution function of a multivariate Normal distribution with covariance\nmatrix given by\n\n\\[\n\\Gamma_{ij}=\\frac{\\partial^{2}}{\\partial t_{i}\\partial t_{j}}K_{\\mathbf{X}}\\left(\\mathbf{t}\\right)\\mid_{\\mathbf{t}=\\mathbf{0}}\n\\]\nmust be computed. For this task there are accurate methods available\nfor up to 20 dimensions \\citet{Genz93comparisonof}.\n\nIf one intends to deal with vectors of dimension at most 5, corresponding\nto multidimensional marginals of the random field modeled, we consider\nmore convenient to use numerical integration of (\\ref{eq:Saddlepoint}).\nFor higher dimensions it would be better to use the result of \\citet{kolassa2010multivariate}\nin order to avoid difficult and inaccurate integrations.\n\n\n\\subsubsection{Differential entropy}\n\nThis also an important interaction manifestation, often encountered\nin statistical research. Using the shorthand notation of \\ref{eq:shorthand_not},\ndefine $Z\\left(\\mathbf{x}\\right):=\\frac{1}{3!}\\kappa^{j_{1},j_{2},j_{3}}h_{j_{1}j_{2}j_{3}}\\left(\\mathbf{x};\\Gamma\\right)$.\n\\citet{barros_2005_2005} studies an approximation to the differential\nentropy of $\\mathbf{X}$, which utilizes only the first correction\nterm in \\ref{eq:edgeworth_series}:\n\n\\begin{multline}\n\\intop f_{\\mathbf{X}}\\left(\\mathbf{x}\\right)\\log\\left(f_{\\mathbf{X}}\\left(\\mathbf{x}\\right)\\right)d\\mathbf{x}=H\\left(\\phi_{\\Gamma}\\right)-\\intop f_{\\mathbf{X}}\\left(\\mathbf{x}\\right)\\log\\left(\\frac{f_{\\mathbf{X}}\\left(\\mathbf{x}\\right)}{\\phi_{\\Gamma}\\left(\\mathbf{x}\\right)}\\right)d\\mathbf{x}\\\\\n\\approx H\\left(\\phi_{\\Gamma}\\right)-\\int\\phi_{\\Gamma}\\left(\\mathbf{x}\\right)\\left(1+Z\\left(\\mathbf{x}\\right)\\right)\\log\\left(1+Z\\left(\\mathbf{x}\\right)\\right)d\\mathbf{x}\\\\\n\\approx H\\left(\\phi_{\\Gamma}\\right)-\\int\\phi_{\\Gamma}\\left(\\mathbf{x}\\right)\\left(Z\\left(\\mathbf{x}\\right)+\\frac{1}{2}Z\\left(\\mathbf{x}\\right)^{2}\\right)d\\mathbf{x}=H\\left(\\phi_{\\Gamma}\\right)-\\frac{1}{12}\\Big\\{\\sum_{j=1}^{J}\\left(k^{j,j,j}\\right)^{2}\\\\\n+3\\sum_{i,j=1,i\\neq j}^{J}\\left(\\kappa^{i,i,j}\\right)^{2}+\\frac{1}{6}\\sum_{i,j,k=1,i1}=0$. In order to avoid identifiability problems\nof the covariance matrix, we set $c_{1}=1$ and declare $\\Gamma$\nto be a true covariance matrix. This model is treated in detail at\n\\citet{ellipticalSpatialRodriguezBardossy}, in the context of spatial\nstatistics; it is shown at \\citet{ellipticalSpatialRodriguezBardossy}\nthat it covers a span of tail dependence going from zero (i.e. Normal)\nto that of the Student-t.\n\n\n\\subsection{Some data}\n\nIn figure \\ref{fig:dataset_Y} an 8-dimensional dataset is presented,\nwith a size of $n=10950$ realization. This dataset may represent\nthe daily (log) return of 8 stocks, or they could represent some daily\nmeasured environmental variable at 8 locations, possibly after transformation.\nIn either case this dataset would amount to a 30 year record. A plot\nof the data appears in figure \\ref{fig:dataset_Y}. We are interested\nin fitting a model that recovers properly the distribution of the\nsum of the components of the 8-dimensional random vector, $S_{\\mathbf{X}}=\\sum_{i=1}^{8}X_{i}$. \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.75\\textwidth]{fig_01}\n\\par\\end{centering}\n\n\\caption{\\label{fig:dataset_Y}8-dimensional test data set}\n\n\n\\end{figure}\n\n\nWe shall employ the model given by c.g.f. (\\ref{eq:archetypal_cgf-1}),\ndue to the shape of data, and to the flexibility of the mentioned\nmodel to represent tail dependence. Specifically, we are interested\nin fitting a model that captures the correlation among the 8 components\nproperly, but \\emph{additionally} provides a good estimation to the\ndistribution of interaction manifestation \n\\begin{equation}\nS_{\\mathbf{X}}=\\sum_{j=1}^{8}X_{j}\n\\end{equation}\n\n\nWe assume for simplicity a mean vector $\\mathbf{m}=\\left(0,\\ldots,0\\right)$\nof zeros (otherwise, data could be standardized to have zero means,\nfirst). As in section \\ref{sub:Connection-of-dependence-sumas}, we\nhave that the c.g.f. of $S_{\\mathbf{X}}$ is given by\n\\begin{equation}\nK_{S_{\\mathbf{X}}}\\left(t\\right)=K_{\\mathbf{X}}\\left(g\\left(t\\right)\\right)=\\frac{c_{1}}{1!}\\left(\\frac{1}{2}g\\left(t\\right)\\Gamma g\\left(t\\right)^{T}\\right)+\\frac{c_{2}}{2!}\\left(\\frac{1}{2}g\\left(t\\right)\\Gamma g\\left(t\\right)^{T}\\right)^{2}+\\frac{c_{3}}{3!}\\left(\\frac{1}{2}g\\left(t\\right)\\Gamma g\\left(t\\right)^{T}\\right)^{3}+\\ldots\\label{eq:CGF_Suma}\n\\end{equation}\nwhere \n\\begin{equation}\ng\\left(t\\right)=\\underbrace{\\left(t,\\ldots,t\\right)}_{8}\n\\end{equation}\n\n\n\n\\subsection{\\label{sub:Parameter-estimation}Parameter estimation}\n\nOur estimating strategy consists of:\n\n\n\\paragraph*{Step 1: Estimate Covariance matrix $\\Gamma$}\n\nIn this way capture much of the 8-dimensional dependence structure.\nSince our model is a member of the elliptical family, we can use the\nestimator for the correlation matrix which uses Kendall's $\\tau$\ncorrelation coefficient (see \\citet{muller_kendalls_2003}),\n\\begin{equation}\n\\hat{cor}\\left(X_{i},X_{j}\\right)=\\sin\\left(\\frac{\\pi}{2}\\tau\\left(X_{i},X_{j}\\right)\\right)\n\\end{equation}\nwhereby a complete correlation matrix, $\\hat{R}$, is obtained.\n\nThen the covariance matrix estimate can be found by \n\\begin{equation}\n\\hat{\\Gamma}=\\Sigma^{\\frac{1}{2}}\\hat{R}\\Sigma^{\\frac{1}{2}}\n\\end{equation}\nwith \n\\[\n\\Sigma=\\left(\\begin{array}{ccc}\nS^{2}\\left(X_{1}\\right) & \\ldots & 0\\\\\n\\vdots & \\ddots & \\vdots\\\\\n0 & \\ldots & S^{2}\\left(X_{8}\\right)\n\\end{array}\\right)\n\\]\nand $S^{2}\\left(X_{j}\\right)$ stands for the sample variance of $X_{j}$.\nThis procedure was followed, resulting in the covariance matrix given\nin table \\ref{tab:Estimated-covariance-for}, at the appendix.\n\nAlternatively, if data represents an environmental variable sampled\nat several locations, standard geostatistical tools can be used to\nestimate $\\Gamma$ (see \\citet{ellipticalSpatialRodriguezBardossy}).\nThe covariance matrix will be in the following considered as known.\n\n\n\\paragraph*{Step 2: Interaction manifestation fitting}\n\nWe do this in a ``method-of-moments'' fashion (method of cumulants,\nshould we say). The $r$-th order cumulant of $S_{\\mathbf{X}}$, $\\kappa_{r}\\left(S_{\\mathbf{X}}\\right)$,\ncan be found by differentiating (\\ref{eq:CGF_Suma}) $r$ times with\nrespect to $t$, and then setting $t=0$. Performing the necessary\ncomputations, one has for the mean and the variance: \n\\begin{eqnarray}\n\\kappa_{1}\\left(S_{\\mathbf{X}}\\right) & = & 0\\\\\n\\kappa_{2}\\left(S_{\\mathbf{X}}\\right) & = & \\frac{c_{1}}{1!}\\frac{2}{2}\\sum_{i,j=1}^{8}\\Gamma_{ij}\n\\end{eqnarray}\nand in general, odd-ordered cumulants will be zero, while even-ordered\ncumulants are given by\n\\begin{equation}\n\\kappa_{2r}\\left(S_{\\mathbf{X}}\\right)=\\frac{c_{r}}{r!}\\frac{\\left(2r\\right)!}{2^{r}}\\left(\\sum_{i_{1},\\ldots,i_{r}=1}^{8}\\sum_{j_{1},\\ldots,j_{r}=1}^{8}\\Gamma_{i_{1}j_{1}}\\ldots\\Gamma_{i_{r}j_{r}}\\right)\\label{eq:cumulantes_de_suma}\n\\end{equation}\n\n\nWe compute the sample cumulants, $\\hat{\\kappa}_{2r}$ (for $r=1,2,3$),\nof $S_{\\mathbf{X}}$. These are found to be 37.426, 463.509 and 105098.112,\nrespectively. Substituting these sample cumulants for the theoretical\ncumulants in (\\ref{eq:cumulantes_de_suma}), and using the already\navailable covariance matrix, $\\Gamma$, we can estimate $c_{1}$,\n$c_{2}$ and $c_{3}$. These estimates are given by $\\hat{c}_{1}=0.999$,\n$\\hat{c}_{2}=0.1101$ and $\\hat{c}_{3}=0.1332$. Note that by considering\ncumulants of $S_{\\mathbf{X}}$ of order $\\geq4$, we can capture important\ntail characteristics of its distribution.\n\n\n\\subsection{Evaluation of the fit}\n\nWe use the Monte Carlo approach to evaluate the fit carried out in\nthe previous sub-section. One can sample from a random vector, $\\mathbf{Y}\\in\\mathbb{R}^{8}$,\nhaving c.g.f. as in (\\ref{eq:archetypal_cgf-1}), by sampling two\nindependent random variables: 1. a non-negative random variable $V>0$,\nwith cumulants $c_{1},\\ldots,c_{r}$ (in our case, $r=3)$; 2. a normally\ndistributed random vector $\\mathbf{X}\\sim N\\left(\\mathbf{0},\\Gamma\\right)$.\nThen one sets: \n\\begin{equation}\n\\mathbf{Y}=\\mathbf{m}+\\sqrt{V}\\times\\mathbf{X}\\label{eq:construccion_Kano}\n\\end{equation}\n\n\nFor more details, the reader is referred to \\citet{ellipticalSpatialRodriguezBardossy}. \n\nWe fitted $V$ as a mixture of 5 gamma random variables, in such a\nway that the cumulants of this mixture are $\\hat{c}_{1}=0.999$, $\\hat{c}_{2}=0.1101$\nand $\\hat{c}_{3}=0.1332$, up to a small error. Then we were able\nto simulate 1000 samples of $\\mathbf{Y}$, each of size $n=10950$,\nusing the fitted parameters. One of the realizations is shown in figure\n\\ref{fig:One-sample-of-Y-new}. Note that the covariance structure\nis mostly recovered, though there are some outliers of a magnitude\nsomewhat larger than those displayed in figure \\ref{fig:dataset_Y}.\nThis is because, once we fitted covariance matrix $\\Gamma$, we focus\non recovering the distribution of the sum of the components of the\nvector $\\mathbf{X}$, i.e. $S_{\\mathbf{X}}$. The outliers there presented\nare part of the mechanism that helps recover the distribution of the\ncomponents sum.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.75\\textwidth]{fig_02}\n\\par\\end{centering}\n\n\\caption{\\label{fig:One-sample-of-Y-new}One sample of size n=10950, generated\nusing the parameters fitted in section \\ref{sub:Parameter-estimation}.}\n\n\n\\end{figure}\n\n\nTo see how well the fitted parameters reproduce $S_{\\mathbf{X}}$,\nwe present several sample quantiles of it, together with confidence\nbands built out of the 1000 Monte Carlo simulations. See table \\ref{tab:Representative-quantiles-of}.\nWe see an excellent cover of the given quantiles, particularly at\nthe tails of the distribution of $S_{\\mathbf{X}}$. \n\n\\begin{table}\n\\begin{centering}\n\\begin{tabular}{|c||c||c||c|}\n\\hline \nQuantile (\\%) & 2.75\\% & 97.5\\% & Observed\\tabularnewline\n\\hline \n\\hline \n0 (min) & -41.921 & -22.644 & -29.191\\tabularnewline\n\\hline \n\\hline \n0.1 & -21.549 & -18.876 & -20.72\\tabularnewline\n\\hline \n\\hline \n0.5 & -16.956 & -15.72 & -17.032\\tabularnewline\n\\hline \n\\hline \n1 & -15.033 & -14.124 & -14.771\\tabularnewline\n\\hline \n\\hline \n5 & -10.248 & -9.748 & -10.049\\tabularnewline\n\\hline \n\\hline \n10 & -7.897 & -7.518 & -7.774\\tabularnewline\n\\hline \n\\hline \n20 & -5.159 & -4.838 & -5.194\\tabularnewline\n\\hline \n\\hline \n25 & -4.138 & -3.838 & -4.212\\tabularnewline\n\\hline \n\\hline \n50 & -0.136 & 0.137 & -0.18\\tabularnewline\n\\hline \n\\hline \n75 & 3.836 & 4.145 & 4.013\\tabularnewline\n\\hline \n\\hline \n80 & 4.84 & 5.155 & 4.987\\tabularnewline\n\\hline \n\\hline \n90 & 7.507 & 7.903 & 7.573\\tabularnewline\n\\hline \n\\hline \n95 & 9.765 & 10.273 & 9.911\\tabularnewline\n\\hline \n\\hline \n99 & 14.114 & 15.058 & 14.293\\tabularnewline\n\\hline \n\\hline \n99.5 & 15.718 & 17.034 & 16.01\\tabularnewline\n\\hline \n\\hline \n99.9 & 18.93 & 21.625 & 20.159\\tabularnewline\n\\hline \n\\hline \n99.99 & 21.908 & 29.07 & 28.542\\tabularnewline\n\\hline \n\\hline \n100 (max) & 22.897 & 43.735 & 28.983\\tabularnewline\n\\hline \n\\end{tabular}\n\\par\\end{centering}\n\n\\caption{\\label{tab:Representative-quantiles-of}Representative quantiles of\n$S_{\\mathbf{X}}$ and confidence bands of 1000 Monte Carlo simulations\nof 10950 sized samples each. The parameters fitted in section \\ref{sub:Parameter-estimation}\nhave been used for the simulation. Simulations reproduce quantiles\nvery similar to those observed.}\n\n\n\\end{table}\n\n\nAdditionally, the distribution of the 365-block maxima of the components\nsums is also acceptably recovered. In figure \\ref{fig:Empirical-Cumulative-Distributio-maxima}\nwe show the empirical distribution function of the 30 sample 365-block\nmaxima (i.e. yearly maxima). The Monte Carlo based 95\\% confidence\nbands for the 365-block maxima of $S_{\\mathbf{X}}$ are also presented\nin figure \\ref{fig:Empirical-Cumulative-Distributio-maxima}.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.75\\textwidth]{fig_03}\n\\par\\end{centering}\n\n\\caption{\\label{fig:Empirical-Cumulative-Distributio-maxima}Empirical Cumulative\nDistribution Function of the 365-block maxima, made out of the 10950\nsized sample presented at figure \\ref{fig:dataset_Y}. Monte Carlo\nsimulation based 95\\% confidence bands have been added from data simulated\nusing the parameters fitted in this section.}\n\n\n\\end{figure}\n\n\n\n\\subsection{More complicated questions}\n\nThe techniques presented in this section can also be used to investigate\nmore complex situations. For example, one would like to model jointly\nthe random variables\n\\begin{eqnarray}\nZ_{1} & := & X_{1}+\\ldots+X_{4}\\\\\nZ_{2} & := & X_{5}+\\ldots+X_{8}\n\\end{eqnarray}\n\n\nThis may be the case if each group of components, $X_{1},\\ldots,X_{4}$\nand $X_{5},\\ldots,X_{8}$, refers each to a geographical area (in\nenvironmental modeling); or if there is some economical reason to\ngroup them (stock price modeling). We may then wish to model the distributions\nof $Z_{1}$ and $Z_{2}$, but also model properly at least the correlation\nbetween them.\n\nApplying a similar computation as before, we find that\n\\begin{equation}\ncov\\left(Z_{1},Z_{2}\\right)=\\frac{c_{1}}{2}\\sum_{i=1}^{4}\\sum_{j=5}^{8}\\Gamma_{ij}\\label{eq:cov_ZZ}\n\\end{equation}\nfor the covariance. Regarding each $Z_{j}$, all odd-ordered cumulants\nare zero, whereas all even-ordered cumulants are given by\n\\begin{equation}\n\\kappa_{2r}\\left(Z_{j}\\right)=\\left(2r-1\\right)!\\times c_{r}\\times\\left(\\frac{R_{j}}{2}\\right)^{r}\\label{eq:cums_ZZ}\n\\end{equation}\nfor $j=1,2$, where\n\\begin{eqnarray*}\nR_{1} & = & 2\\sum_{i=1}^{4}\\Gamma_{ii}+4\\sum_{11}=0$ as in the original model by \\citet{sanso_venezuelan_1999}.\nAs shown by \\citet{ellipticalSpatialRodriguezBardossy}, a random\nfield with $\\left(c_{1},\\ldots,c_{5}\\right)$ as above is practically\nindistinguishable in its one and two dimensional marginal distributions\nfrom a Gaussian field with the same covariance function and mean.\nHowever, implications for the interaction manifestation ``average\nof fields components'', where each component represents daily precipitation\nover an 500 mt $\\times$ 500 mt squared area on the Saalach river\ncatchment, are significant. \n\nThe authors obtained 3000 conditional simulations, given the rainfall\ndata available, of the rainfall field over the Saalach river catchment\nfor June 1st 2013, a day of intense rainfall during the 2013 central\nEuropean floods. In figure \\ref{fig:Two-conditionally-simulated-fields},\ntwo of the obtained conditional fields are presented, using the Gaussian\nand the almost-Gaussian latent structure. In figure \\ref{fig:Boxplots-of-the-conditional},\nwe show the distribution of the conditional values of mean precipitation\nover the catchment, for both latent structures. Note that the multivariate\ninteractions, hardly noticeable on the one and two dimensional marginal\ndistributions, increase dramatically the probability of a very high\nmean precipitation over the studied catchment. The consequence is\nthat substantial under-estimation of flood return periods may me incurred,\nif one does not account for interaction among more than tow components,\nin one's spatio-temporal precipitation models.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.5\\textwidth]{fig_04}\\includegraphics[width=0.5\\textwidth]{fig_05}\n\\par\\end{centering}\n\n\\caption{\\label{fig:Two-conditionally-simulated-fields}Two conditionally simulated\nfields for June 1st 2013, for part of the Saalach river catchment: Field with Gaussian latent structure (left),\nand field with non-Gaussian latent structure (right). Stations providing\nthe observed data are indicated in red. Stations indicated by blue\npoints have no available data for that day. Note the intense precipitation\nclusters predictable by the model with latent field having multivariate\ninteractions.}\n\n\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.75\\textwidth]{fig_06}\n\\par\\end{centering}\n\n\\caption{\\label{fig:Boxplots-of-the-conditional}Boxplots of the average of\nthe conditionally simulated random fields for June 1st 2013, in millimeters,\nfor the Saalach river catchment. The field with high oder interacting\nlatent structure shows much more variability. In particular, average\nprecipitation over the catchment above 120 mm are quite probable under\nthis model.}\n\n\n\\end{figure}\n\n\n\n\\section{\\label{sec:Discussion}Discussion}\n\nAn approach for considering interactions that go beyond correlations\nhas been presented. We have seen that the discrimination between interactions\n``parameters'' and interactions ``manifestations'' can help to\ncircumvent two major problems one is confronted with when attempting\nto quantify and model higher order interactions: the problem of interpretability,\nby working with subject-matter relevant manifestations of interdependence;\nand the problem of high dimensionality, by recoursing to joint cumulants\nas building blocks of a dependence model. By using the cumulant generating\nfunction, we are recoursing to a well-studied object: the characteristic\nfunction of a distribution. \n\nAs dimension of vector $\\mathbf{X}$ increases, interactions of high\norder may be more and more difficult to assess. For example, a random\nvector having c.f.g. (\\ref{eq:archetypal_cgf-1}), with $c_{1}=1$\n, $c_{r}\\approx0$ for $2\\leq r\\leq3$ but then $c_{r\\geq4}\\neq0$,\nwould have one and two dimensional marginals practically equal to\nthose of a Guassian distribution. But the interaction coefficients\nof groups of 14 components or more will be very different, producing\nvery different interaction manifestations. The difference in the overall\ndependence structures may grow tremendously as the dimension of the\nrandom vector $\\mathbf{X}$ grow (i.e. $J>>2$), even though these\nfact may go totally unnoticed in the one and two dimensional marginal\nanalysis of data. \n\nIn \\citet{ellipticalSpatialRodriguezBardossy}, these issues are dealt\nwith and illustrated in the context of Spatial Statistics, where the\nissue of low dimensionality is essential, and where interaction manifestations\ncan differ drastically between two models having very similar 1 and\n2 dimensional marginals, due to the big dimension of the field.\n\n\n\\subsection*{Acknowledgments}\n\nThis research forms part of the Ph.D thesis of the first author, which\nwas funded by a scholarship of the German Academic Exchange Service\n(DAAD). This Ph.D work was carried out within the framework of the\nENWAT program at the University of Stuttgart.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{Introduction}}\n\nStrongly correlated electron systems, such as heavy fermion compounds,\nhigh-temperature superconductors, have gained much attention from both\ntheoretical and experimental point of view. The competition between the\nkinetic energy and strong Coulomb interaction of fermions generates a lot\nof fascinating phenomena. Various theoretical approaches have been developed\nto treat the regime of intermediate coupling. The widely used perturbative\nmethods, such as random phase approximation (RPA), fluctuation exchange\n(FLEX)\\cite{Bickers-1989,Bickers-1991}, and the two-particle self-consistent\n(TPSC)\\cite{Tremblay-1995,Tremblay-1994} method are based on the expansion in\nthe Coulomb interaction which is only valid in weak-coupling. To go beyond the\nperturbative approximation and to gain insight of the correlation effects of\nthe fermion systems, new theoretical methods are needed. Dynamical mean field\ntheory (DMFT)\\cite{Metzner-1989,Hartmann-1984,George-1996} is a big step\nforward in the understanding Metal-Insulator transition. \n\nDynamical mean field theory maps a many-body interacting system on a lattice\nonto a single impurity embedded in a non-interacting bath. Such a mapping\nbecomes exact in the limit of infinite coordination number. All local temporal\nfluctuations are taken into account in this theory, however spatial\nfluctuations are treated on the mean field level. DMFT has been proven a\nsuccessful theory describing the basic physics of the Mott-Hubbard\ntransition. But the non-local correlation effect can't always be\nomitted. Although, straight forward extensions of\nDMFT\\cite{Hettler-1998,Kotliar-2001,Okamoto-2003,Potthoff-2003,Maier-2005} \nhave captured the influence of short-range correlation, these methods are\nstill not capable of describing the collective behavior, e.g. spin wave\nexcitations of\nmany-body system. At the same time, most of the numerically exact impurity\nsolvers require a substantial amount of time to achieve a desired accuracy even\non a small cluster, which makes the investigation of larger lattice to be\nimpossible. \n\nRecently, some efforts have been made to take the spatial fluctuations into\naccount in different ways\\cite{Toschi-2007, Rubtsov-2006, Kusunose-2006,\n Tokar-2007, Slezak-2006}. All these methods construct the non-local\ncontribution of DMFT from the local two-particle vertex. The electron\nself-energy is expressed as a function of the two-particle vertex and \nthe single-particle propagator. The cluster extention of DMFT considers\nthe correlation within the small cluster. Compared to these, the \ndiagrammatic re-summation technique involved in these new methods makes them\nonly approximately include the non-local corrections. While, long range\ncorrelations are also considered in these methods and the computational burden\nis not serious. \n\nIn this paper we will apply the method of Rubtsov\\cite{Rubtsov-2006} to\nconsider the vertex renormalization of the DF through the Bethe-Salpeter\nequation. Lattice susceptibility is calculated from the renormalized DF\nvertex. \n\nThe paper is organized as follows: In Sec. \\ref{Details} we summarize the\nbasic idea of the DF method and give details of the calculation. The\nDMFT two-particle Green's function and the corresponding vertex calculation\nare implemented in CT-QMC in Sec. \\ref{Vertex}. The frequency dependent vertex\nis modified through the Bethe-Salpeter Equation to obtain the momentum\ndependence in Sec. \\ref{Mom-Vex}. In Sec. \\ref{application} we present the\ncalculation of the lattice susceptibility and compare it with QMC results and\nalso the works from Toschi\\cite{Toschi-2007}. The conclusions are summarized\nin Sec. \\ref{conclusion}, where we also present possible application.\n\n\\section{The DF method\\label{Details}}\n\nWe study the general one-band Hubbard model at two dimensions\n\\begin{equation}\n H=\\sum_{k,\\sigma}\\epsilon_{k,\\sigma}c_{k\\sigma}^{\\dagger}\n c_{k\\sigma}^{\\phantom\\dagger}+U\\sum_{i}n_{i\\uparrow}n_{i\\downarrow} \n\\end{equation}\n$c_{k\\sigma}^{\\dagger}(c_{k\\sigma})$ creates (annihilates) an electron with\nspin-$\\sigma$ and momentum $k$. The dispersion relation is\n$\\epsilon_{k}=-2t\\sum_{i=1}^{N}\\cos k_{i}$, where $N$ is the number of lattice\nsites. The basic idea of the DF method\\cite{Rubtsov-2006} is to\ntransform the hopping between different sites into coupling to an auxiliary \nfield $f(f^{\\dagger})$. By doing so, each lattice site can be viewed as an\nisolated impurity. The interacting lattice problem is reduced to solving a\nmulti-impurity problem which couples to the auxiliary field. This can be done\nusing the standard DMFT calculation. After integrating out the lattice\nfermions $c(c^{\\dagger})$ one can obtain an effective theory of the auxiliary\nfield where DMFT serves as an starting point of the expansion over the\ncoupling between each impurity site with the auxiliary field. \n\nTo explicitly demonstrate the above idea we start from the action of DMFT\nwhich can be written as \n\\begin{equation}\\label{original_fermion}\n S[c^{+},c]=\\sum_{i}S_{imp}^{i}-\\sum_{\\nu,k,\\sigma}(\\Delta_{\\nu}\n -\\epsilon_{k\\nu})c_{\\nu k\\sigma}^{\\dagger}c_{\\nu k\\sigma}^{\\phantom\\dagger}\n\\end{equation} \nwhere $\\Delta_{\\nu}$ is the hybridization function of the impurity problem\ndefined by $S_{imp}^{i}$ which is the action of an isolated impurity at site\n$i$ with the local Green's function $g_{\\nu}$. Using the Gaussian identity, we\ndecouple the lattice sites into many impurities which couple only to the field\n$f$ \n\\begin{eqnarray}\\label{auxiliary_field}\n S[c^{\\dagger},c;f^{\\dagger},f]&=&\\sum_{i}S_{imp}^{i}+\\sum_{k,\\nu,\\sigma}\n [g_{\\nu}^{-1}(c_{k\\nu\\sigma}^{\\dagger}f_{k\\nu\\sigma}^{\\phantom\\dagger}+h.c.)\n \\nonumber\\\\\n &&\\hspace{1cm}+g_{\\nu}^{-2}(\\Delta_{\\nu}-\\epsilon_{k})^{-1}\n f_{k\\nu\\sigma}^{\\dagger}f_{k\\nu\\sigma}^{\\phantom\\dagger}] \n\\end{eqnarray}\nThe equivalence of Eqs. (\\ref{original_fermion}) and (\\ref{auxiliary_field})\nform an exact relation between the Green's funtion of the lattice electrons and\nthe DF. \n\\begin{equation}\\label{relation}\n G_{\\nu,k}=g_{\\nu}^{-2}(\\Delta_{\\nu}-\\epsilon_{k})^{-2}\n G_{\\nu,k}^{d}+(\\Delta_{\\nu}-\\epsilon_{k})^{-1}\n\\end{equation}\nThis relation is easily derived by considering the derivative over\n$\\epsilon_{k}$ in the two actions. Eq. (\\ref{relation}) allows now to solve\nthe many-body ``lattice'' problem based on DMFT which is different from the\nstraight forward cluster extension. The problem is now to solve the Green's\nfunction of the DF $G^{d}_{\\nu,k}$. It is determined by integrating \nEq. (\\ref{auxiliary_field}) over $c^{\\dagger}$ and $c$ yielding a Taylor\nexpansion series in powers of $f^{\\dagger}$ and $f$. The Grassmann integral\nensures that $\\bar{f}$ and $f$ appear only in pairs associated with the\nlattice fermion n-particle vertex obtained from the single-site DMFT\ncalculation. In this paper we restrict our considerations to the two-particle\nvertex $\\gamma^{(4)}$.\n \n\\begin{figure}[b]\n \\includegraphics[width=200pt]{Self-Energy}%\n \\caption{The first two self-energy diagrams. They are composed of the local\n vertices function and DF propagator. \\label{Self-Energy}}\n\\end{figure}\n\nExpanding the Luttinger-Ward functional in $\\gamma^{4}$, the first two\ncontributions to the self energy function are the diagrams shown in\nFig. \\ref{Self-Energy}. Diagram (a) vanishes for the bare DF since this\ndiagram exactly corresponds to the DMFT self consistency. Therefore the first\nnon-local contribution is given by diagram (b). The self-energy for these two\ndiagrams are \n\\begin{subequations}\n \\begin{align}\n \\Sigma^{(1)}_{\\sigma}(k_{1}) &= -\\frac{T}{N}\\sum_{\\sigma^{\\prime},\n k_{2}}G_{\\sigma^{\\prime}}^{d}\n (k_{2})\\gamma^{(4)}_{\\sigma\\sigma^{\\prime}}\n (\\nu,\\nu^{\\prime};\\nu^{\\prime},\\nu) \\label{self-energy1} \\\\\n \\Sigma^{(2)}_{\\sigma}(k_{1}) &= -\\frac{T^{2}}{2N^{2}}\\sum_{2,3,4}\n G^{d}_{\\sigma_{2}}(k_{2})G^{d}_{\\sigma_{3}}(k_{3})\n G^{d}_{\\sigma_{4}}(k_{4})\\nonumber\\\\\n & \\gamma^{(4)}_{\\sigma_{1234}}(\\nu_{1},\\nu_{2};\\nu_{3},\\nu_{4})\n \\gamma^{(4)}_{\\sigma_{4321}}(\\nu_{4},\\nu_{3};\\nu_{2},\\nu_{1})\\nonumber\\\\\n & \\delta_{k_{1}+k_{2}, k_{3}+k_{4}}\\delta_{\\sigma_{1}+\\sigma_{2},\n \\sigma_{3}+\\sigma_{4}} \\label{self-energy2}\n \\end{align}\n\\end{subequations}\nHere space-time notation is used, $k=(\\vec{k},\\nu)$,\n$q=(\\vec{q},\\omega)$. Fermionic Matsubara frequency is \n$\\nu_{n}=(2n+1)\\pi\/\\beta$, bosonic frequency is $\\omega_{m} = 2m\\pi\/\\beta$\nwhere $\\beta$ is the inverse temperature. Together with the bare DF\nGreen's function\n$G^{d}_{0}(k)=-g_{\\nu}^{2}\/[(\\Delta_{\\nu}-\\epsilon_{k})^{-1}+g_{\\nu}]$, the\nnew Green's function can be derived from the Dyson equation \n\\begin{equation}\\label{Dyson}\n [G^{d}(k)]^{-1}=[G_{0}^{d}(k)]^{-1}-\\Sigma^{d}(k)\n\\end{equation}\n\nThe algorithm of the whole calculation is: \n\\begin{enumerate}\n\\item Set initial value of $\\Delta_{\\nu}$ for the first DMFT loop.\n\\item Determine the single-site DMFT Green's function $g_{\\nu}$ from\n the hybridization function $\\Delta_{\\nu}$. The self-consistency condition\n ensures that the first diagram of the DF self-energy is very small. \n\\item Go through the DMFT loop once again to calculate the two-particle\n Green's function and corresponding $\\gamma$-function. The method for\n determining the $\\gamma$-function is implemented for both strong and\n weak-coupling CT-QMC in the next section of this paper. \n\\item Start an inner loop calculation to determine the DF Green's\n function and in the end the lattice Green's function.\n \\begin{enumerate}\n \\item From Eqs. (\\ref{self-energy1}), (\\ref{self-energy2}) and the Dyson\n equation (\\ref{Dyson}) to calculate the self-energy of the DF. \n \\item Repeatly use Eq. (\\ref{self-energy1}), (\\ref{self-energy2}) and\n Eq. (\\ref{Dyson}) until the convergence of the DF Green's function\n is achived. \n \\item The lattice Green's function is then given by Eq. (\\ref{relation})\n from that of the DF.\n \\end{enumerate}\n\\item Fourier transform the momentum lattice Green's function into real space.\n And from the on-site component $G_{ii}$ to determine a new hybridization\n function $\\Delta_{\\nu}$ which is given by Eq. (\\ref{Old-New}).\n\\item Go back to the Step 3. and iteratively perform the outer loop until the\n hybridization $\\Delta_{\\nu}$ doesn't change any more. \n\\end{enumerate} \n\nAlthough diagram (a) is exactly zero for the bare DF Green's\nfunction, it gives non-zero contribution from the second loop where the DF\nGreen's function is updated from Eq. (\\ref{Dyson}). As a result, the \nhybridization function should also be updated before the next DMFT loop is\nperformed . This is simply done by setting the local full DF Green's\nfunction to zero, together with the condition that the old hybrization\nfunction forces the bare local DF Green's function to be zero\n($\\sum_{k}G^{0,d}_{\\nu,k}=0$), we obtain a set of equations \n\\begin{subequations}\n \\begin{align}\n & \\frac{1}{N}\\sum_{k}[G_{\\nu,k} - (\\Delta^{New}_{\\nu}-\\epsilon_{k})^{-1}]\n g_{\\nu}^{2}(\\Delta^{New}_{\\nu}-\\epsilon_{k})^{2} = 0 \\\\\n & \\frac{1}{N}\\sum_{k}[G^{0}_{\\nu,k} -\n (\\Delta^{Old}_{\\nu}-\\epsilon_{k})^{-1}] \n g_{\\nu}^{2}(\\Delta^{Old}_{\\nu}-\\epsilon_{k})^{2} = 0 \n \\end{align}\n\\end{subequations}\nwhich yields \n\\begin{equation}\n \\Delta_{\\nu}^{New}-\\Delta_{\\nu}^{Old}\\approx\\frac{1}{N}\\sum_{k}(G_{\\nu,k}-\n G^{0}_{\\nu,k})(\\Delta^{Old}_{\\nu}-\\epsilon_{k})^{2}\n\\end{equation}\nThis equation finally gives us the relation between the new and old\nhybridization function. \n\\begin{equation}\\label{Old-New}\n \\Delta_{\\nu}^{New}=\\Delta_{\\nu}^{Old}+g_{\\nu}^{2}G_{loc}^{d}\n\\end{equation}\n\nIn the whole calculation, the DF perturbation calculation converges\nquickly. The most time consuming part of this method is the DMFT calculation\nof the two particle Green's function. There are some useful symmetries to\naccelerate the calculation. As already pointed out\\cite{Abrikosov:QFT,\n Nozieres:1964}, it is convenient to take the symmetric form of the\ninteraction term. The two particle Green's function is then a fully\nantisymmetric function. Such fully antisymmetric form is very useful to speed\nup the calculation of the two particle Green's function. One does not need to\ncalculate all the frequency points within the cutoff in Mastsubara space, a\nfew special points are calculated and the values for the other points are\ngiven by that of those special points through antisymmetric property. In the\nDF self energy calculation, we always have the convolution type of momentum\nsummation which is very easy to be calculated by fast fourier transform (FFT). \n\n\\section{CT-QMC and two-particle vertex}\\label{Vertex}\n\nFrom the above analysis, the key idea of the DF method is to\nconstruct the nonlocal contribution from the auxiliary field and the DMFT\ntwo-particle Green's function. Therefore it is quite important to accurately \ndetermine the two-particle vertex. Here we adapt the newly developed CT-QMC\nmethod\\cite{Rubtsov-2005, Werner-2006(1), Werner-2006(2)} to calculate the two\nparticle Green's function $\\chi$. \n\nFirst we briefly outline the CT-QMC technique. For more details, we refer the\nreaders to\\cite{Rubtsov-2005, Werner-2006(1), Werner-2006(2)}. Here we discuss\nthe two-particle Green's function and some numerical implemetations in more\ndetailed. Two variants of the CT-QMC methods have been proposed based on the\ndiagrammatic expansion. Unlike the Hirsch-Fye method, these methods don't have\na Trotter error and can approach the low temperature region easily. In the\nweak-coupling method\\cite{Rubtsov-2005} the non-interacting part of the\npartition function is kept and expanded the interaction term into Taylor\nseries. Wick's theorem ensures that the corresponding expansion can be written\ninto a determinant at each order \n\\begin{equation}\n {\\cal Z}=\\sum_{k}\\frac{(-U)^{k}}{k!}\\int d\\tau_{1}\\cdots d\\tau_{k}e^{-S_{0}}\n \\det[D_{\\uparrow}D_{\\downarrow}]\n\\end{equation} \nwith\n\\begin{equation}\n D_{\\uparrow}D_{\\downarrow}=\n \\left(\n \\begin{array}{cc}\n \\cdots & G_{\\uparrow}(\\tau_{1}-\\tau_{k}) \\cr\n \\cdots & \\cdots \n \\end{array}\n \\right)\n \\left(\n \\begin{array}{cc}\n \\cdots & \\cdots \\cr\n G_{\\downarrow}(\\tau_{k}-\\tau_{1}) & \\cdots\n \\end{array}\n \\right)\n\\end{equation}\nwhere $S_{0}$ is the non-interacting action and $G^{0}$ is the Weiss field, \nand the one-particle Green's function is measused as \n\\begin{equation}\n G(\\nu)=G^{0}(\\nu)-\\frac{1}{\\beta}G^{0}(\\nu)\\sum_{i,j}M_{i,j}\n e^{i\\nu(\\tau_{i}-\\tau_{j})}G^{0}(\\nu)\n\\end{equation}\n\nIn the strong coupling method the effective action is expanded in the\nhybridization function by integrating over the non-interacting bath degrees of\nfreedom. Such an expansion also yields a determinant. \n\\begin{eqnarray}\n &&{\\cal Z}=TrT_{\\tau}e^{-S_{loc}}\\prod_{\\sigma}\n \\sum_{k_{\\sigma}}\\frac{1}{k_{\\sigma}!}\\int d\\tau_{1}^{s}\\cdots\n d\\tau_{k_{\\sigma}}^{s}\\int d\\tau_{1}^{e}\\cdots d\\tau_{k_{\\sigma}}^{e}\n \\nonumber\\\\\n &&\\Psi_{\\sigma}(\\tau^{e})\\left(\n \\begin{array}{ccc}\n \\Delta(\\tau_{1}^{e}-\\tau_{1}^{s}) & \\cdots & \\Delta(\\tau_{1}^{e}\n -\\tau_{k_{\\sigma}}^{s})\\\\\n \\cdots & \\ddots & \\cdots \\\\\n \\Delta(\\tau_{k_{\\sigma}}^{e}-\\tau_{1}^{s}) & \\cdots & \n \\Delta(\\tau_{k_{\\sigma}}^{e}-\\tau_{k_{\\sigma}}^{s})\n \\end{array}\\right)\n \\Psi_{\\sigma}^{\\dagger}(\\tau^{s})\n\\end{eqnarray}\nHere $\\Psi(\\tau) = (c_{1}(\\tau), c_{2}(\\tau),\\cdots,\nc_{k_{\\sigma}(\\tau)})$. The action is evaluated by a Monte Carlo random walk\nin the space of expansion order $k$. Therefore the corresponding hybridization\nmatrix changes in every Monte Carlo step. One particle Green's function is\nmeasured from the expansion of hybridization function as\n$G(\\tau_{j}^{e}-\\tau_{i}^{s})=M_{i,j}$. $M$ is the inverse matrix of the\nhybridization function. Apparently one needs to calculate this inverse matrix\nin every update step which is time consuming, fortunately it can be obtained by\nthe fast-update algorithm\\cite{Rubtsov-2005}. \n\nAt the same time such a relation allows direct measurement of the Matsubara\nGreen's function \n\\begin{equation}\n G(i\\nu_{n})=\\frac{1}{\\beta}\\sum_{i,j}e^{-i\\nu_{n}\\tau_{i}^{s}}M_{i,j}\n e^{i\\nu_{n}\\tau_{j}^{e}}\n\\end{equation}\n\nCompared with the imaginary time measurement, it seems additional\ncomputational time is needed for the sum over every matrix elements\n$M_{i,j}$. K. Haule proposed to implement such measurement in every fast update\nprocedure which makes sure that only linear amount of time is\nneeded\\cite{Haule-2007}. \n\nIn our calculation the Green's function is measured in the weak-coupling\nCT-QMC at each accepted update which greatly reduces the computational\ntime. The weak-coupling CT-QMC normally yields a higher perturbation order $k$\nthan the strong-coupling CT-QMC. It seems that the performance of the\nstrong-coupling CT-QMC is better\\cite{Emanuel-2007}. Concerning the\nconvergence speed, the weak-coupling CT-QMC is almost same as the\nstrong-coupling one under the above implementation together with a proper\nchoice of $\\alpha$, since in strong-coupling CT-QMC more Monte Carlo steps are \nneeded usually in order to smooth the noise of Green's function at imaginary\ntime around $\\beta\/2$ or at large Matsubara frequency points. Furthermore, the\nweak-coupling CT-QMC is much easier implemented for large cluster DMFT\ncalculation, in which case the strong-coupling method needs to handle a big\neigenspace. In this paper we mainly use weak-coupling CT-QMC as impurity \nsolver, while all the results can be obtained in the strong-coupling CT-QMC\nwhich was used as an accuracy check. \n\nSimilarly, we adapt K. Haule's implementation to calculate the two-particle\nGreen's function in frequency space. In the weak coupling CT-QMC, the\nnon-interacting action has Gaussian form which ensures the applicability of\nWick's theorem for measuring the two particle Green's function \n\\begin{eqnarray}\\label{2PG}\n \\chi_{\\sigma\\sigma^{\\prime}}(\\nu_{1},\\nu_{2},\\nu_{3},\\nu_{4})&=&\n T[\\overline{G_{\\sigma}(\\nu_{1},\\nu_{2})G_{\\sigma^{\\prime}}\n (\\nu_{3},\\nu_{4})}\\nonumber\\\\\n &-&\\delta_{\\sigma\\sigma^{\\prime}}\\overline{\n G_{\\sigma}(\\nu_{1},\\nu_{4})G_{\\sigma}(\\nu_{3},\\nu_{2})}]\n\\end{eqnarray} \nThe over-line indicates the Monte Carlo average. In each Monte Carlo\nmeasurement, $G(\\nu,\\nu^{\\prime})$ depends on two different argument $\\nu$ \nand $\\nu^{\\prime}$, only in the average level,\n$\\overline{G(\\nu,\\nu^{\\prime})}=G(\\nu)\\delta_{\\nu,\\nu^{\\prime}}$ is a function\nof single frequency. In each fast-update procedure, the new and old\n$G(\\nu,\\nu^{\\prime})$ have a closed relation which ensures that one can\ndetermine the updated Green's function $G^{New}(\\nu,\\nu^{\\prime})$ from the\nold one $G^{Old}(\\nu,\\nu^{\\prime})$. For example, adding pair of kinks and \nsupposing before updating the perturbation order is $k$, then it is $k+1$\nfor the new M-matrix. The new inserted pair is at $k+1$ row and $k+1$\ncolumn. \n\\begin{eqnarray}\\label{strong-update}\n &&G^{New}(\\nu,\\nu^{\\prime})-G^{old}(\\nu,\\nu^{\\prime})\\nonumber\\\\\n &=&\\frac{M^{New}_{k+1,k+1}}{\\beta}G^{0}(\\nu)\\left\\{XL\\cdot XR-XR\\cdot\n e^{-i\\nu\\tau_{k+1}^{s}}\\right.\\nonumber\\\\\n &&\\left.\\hspace{0.5cm}-XL\\cdot e^{i\\nu^{\\prime}\\tau_{k+1}^{e}}\n +e^{-i\\nu\\tau_{k+1}^{s}+i\\nu^{\\prime}\\tau_{k+1}^{e}}\\right\\}\n G^{0}(\\nu^{\\prime}) \n\\end{eqnarray}\nHere, $XL=\\sum_{i=1}^{k}e^{-i\\nu\\tau_{i}^{s}}L_{i}$,\n$XR=\\sum_{j=1}^{k}e^{i\\nu^{\\prime}\\tau_{j}^{e}}R_{j}$ and $L_{i}, R_{j}$ have\nthe same definition as in Ref\\cite{Rubtsov-2005}. In every step, one only needs\nto calculates the Green's function when the update is accepted and only a few\ncalculations are needed. A similar procedure for removing pairs, shiftting\nend-point operation can be used. Such method is also applicable in the segment\npicture of strong-coupling CT-QMC. In the weak-coupling CT-QMC, such an\nimplementation greatly improves the calculating speed in low temperature and\nstrong interaction regime\\footnote{In fact, the improvement is more obvious\n for larger M-matrices. The strong coupling CT-QMC and the weak coupling\n CT-QMC require approximately the same amount of CPU time although in the\n weak coupling case the average perturbation order is higher than in the\n strong coupling case}. Once one obtains the two frequency dependent\nGreen's function in every monte carlo step, the two-particle Green's function\ncan be determined easily from Eq. (\\ref{2PG}). The two-particle vertex is then\ngiven from the following equation: \n\\begin{equation}\n \\gamma^{\\sigma\\sigma^{\\prime}}_{\\omega}(\\nu,\\nu^{\\prime})=\n \\frac{\\beta^{2}[\\chi^{\\sigma\\sigma^{\\prime}}_{\\omega}(\\nu,\\nu^{\\prime})\n -\\chi^{0}_{\\omega}(\\nu,\\nu^{\\prime})]}\n {g_{\\sigma}(\\nu)g_{\\sigma}\n (\\nu+\\omega)g_{\\sigma^{\\prime}}(\\nu^{\\prime}+\\omega) \n g_{\\sigma^{\\prime}}(\\nu^{\\prime})}\n\\end{equation}\nwhere \n\\begin{equation}\n \\chi^{0}_{\\omega}(\\nu,\\nu^{\\prime})=T[\\delta_{\\omega,0}g_{\\sigma}(\\nu)\ng_{\\sigma^{\\prime}}(\\nu^{\\prime})-\\delta_{\\sigma\\sigma^{\\prime}}\n\\delta_{\\nu,\\nu^{\\prime}}g_{\\sigma}(\\nu)g_{\\sigma}(\\nu+\\omega)]\n\\end{equation} \nis the bare susceptibility. For the multi-particle Green's function, it still\ncan be constructed from the two frequency dependent Green's function\n$G(\\nu,\\nu^{\\prime})$, but more terms appear from Wicks theorem. Simply, when\nset $\\nu=\\nu^{\\prime}$ one can calculate the one-particle Green's funtion\neasily. \n\n\\section{Momentum dependece of Vertex}\\label{Mom-Vex}\n\nAs mentioned earlier diagram (a) in Fig. \\ref{Self-Energy} only gives the\nlocal contribution. The first non-local correction in the DF method\nis from diagram (b). Momentum dependences comes into this theory through the\nbubble-like diagram between the two vertices which yields the momentum\ndependence of the DF vertex. The natural way to renormalize vertex is\nthrough the Bethe-Salpeter equation. Since the DMFT vertex is only a function\nof Matsubara frequency, the integral over internal momentum $k$ and\n$k^{\\prime}$ ensures that the full vertex only depends on the center of mass\nmomentum $Q$. The Bethe-Salpeter equation in the particle-hole\nchannel\\cite{Abrikosov:QFT, Nozieres:1964} are shown in\nFig. \\ref{BSE-channel}. \n\nFrom the construction of the DF method, we know the interaction of the\nDF is coming from the two particle vertex of lattice fermion which is\nobtained through DMFT calculation. In the Bethe-Salpeter equation, it plays\nthe role of the building-block. The corresponding Bethe-Salpeter equation for\nthese two channels are\n\\begin{subequations}\\label{BSE}\n \\begin{align}\n & \\Gamma^{ph0,\\sigma\\sigma^{\\prime}}_{Q}(\\nu,\\nu^{\\prime}) = \n \\gamma^{\\sigma\\sigma^{\\prime}}_{\\omega}(\\nu,\\nu^{\\prime})- \\nonumber\\\\\n &\\frac{T}{N}\\sum_{k^{\\prime\\prime}\\sigma^{\\prime\\prime}}\n \\gamma^{\\sigma\\sigma^{\\prime\\prime}}_{\\omega}(\\nu,\\nu^{\\prime\\prime})\n G^{d}(k^{\\prime\\prime})G^{d}(k^{\\prime\\prime}+Q)\n \\Gamma^{ph0,\\sigma^{\\prime\\prime}\\sigma^{\\prime}}_{Q}\n (\\nu^{\\prime\\prime},\\nu^{\\prime}) \\\\\n & \\Gamma^{ph1,\\sigma\\bar{\\sigma}}_{Q}(\\nu,\\nu^{\\prime}) = \n \\gamma^{\\sigma\\bar{\\sigma}}_{\\omega}(\\nu,\\nu^{\\prime})- \\nonumber\\\\\n &\\frac{T}{N}\\sum_{k^{\\prime\\prime}}\n \\gamma^{\\sigma\\bar{\\sigma}}_{\\omega}(\\nu,\\nu^{\\prime\\prime})\n G^{d}(k^{\\prime\\prime})G^{d}(k^{\\prime\\prime}+Q)\n \\Gamma^{ph1,\\sigma\\bar{\\sigma}}_{Q}(\\nu^{\\prime\\prime},\\nu^{\\prime})\n \\end{align}\n\\end{subequations} \nHere, the short hand notation of spin configuration is\nused. $\\gamma^{\\sigma\\sigma^{\\prime}}$ represents\n$\\gamma^{\\sigma\\sigma\\sigma^{\\prime}\\sigma^{\\prime}}$, while\n$\\gamma^{\\sigma\\bar{\\sigma}\\bar{\\sigma}\\sigma}$ is denoted by\n$\\gamma^{\\sigma\\bar{\\sigma}}$ where\n$\\bar{\\sigma}=-\\sigma$. $\\Gamma^{ph0(ph1)}$ are the full vertices in the\n$S_{z}=0$ and $S_{z}=\\pm1$ channel, respectively. $G^{d}$ is the full DF\nGreen's function obtained from section \\ref{Details} which is kept unchanged\nin the calculation of the Bethe-Salpeter Equation. Different from the work of\nS. Brener\\cite{Brener-2007}, we solve the above equations directly in momentum\nspace with the advantage that in this way we can calculate the susceptibility\nfor any specific center of mass momentum $Q$ and it's convenient to use FFT for\ninvestigating larger lattice. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=230pt]{BSE-channel}\n \\caption{$S_{z}=0$ (ph0) and $S_{z}=\\pm1$ (ph1) particle-hole channels of\n the DF vertex, between vertices there are two full DF\n Green's function. The $S_{z}=\\pm1$ component is the triplet channel,\n while that for $S_{z}=0$ can be either singlet or triplet.} \n \\label{BSE-channel}\n \\end{center}\n\\end{figure}\n\nIn Eq. (\\ref{BSE}) one has to sum over the internal spin indices in the\n$S_{z}=0$ channel which is not present in $S_{z}=\\pm1$ channel. One can\ndecouple the $S_{z}=0$ channel into the charge and spin channels\n$\\gamma_{c(s)}=\\gamma^{\\sigma\\sigma}\\pm\\gamma^{\\sigma\\bar{\\sigma}}$ which can\nbe solved seperately, and it turns out that the spin channel vertex function \nis exactly same as the that in $S_{z}=\\pm1$ channel, see e.g.\nP. Nozieres\\cite{Nozieres:1964}. Such relation is true for the DMFT vertex,\nand was also verified for the momentum dependent vertex in the DF\nmethod\\cite{Brener-2007}. In our calculation, we have solved the $S_{z}=0$\nchannel by decoupling it to the charge and spin channel, while the $ph1$\nchannel is not used. \n\nOnce the converged momentum dependent DF vertex is obtained, one can\ndetermine the corresponding DF susceptibility in the standard way by\nattaching four Green's functions to the DF vertex. \n\\begin{subequations}\n \\begin{align}\n & \\chi^{\\sigma\\sigma^{\\prime}}_{d}(Q) = \\chi^{0}_{d}(Q)+\\nonumber\\\\\n & \\frac{T^{2}}{N^{2}}\n \\sum_{k,k^{\\prime}}G^{d}_{\\sigma}(k)G^{d}_{\\sigma}(k+Q)\n \\Gamma^{\\sigma\\sigma^{\\prime}}(Q)\n G^{d}_{\\sigma^{\\prime}}(k^{\\prime})G^{d}_{\\sigma^{\\prime}}(k^{\\prime}+Q) \\\\\n & \\chi^{\\sigma\\bar{\\sigma}}_{d}(Q) = \\chi^{0}_{d}(Q)+\\nonumber\\\\\n & \\frac{T^{2}}{N^{2}}\n \\sum_{k,k^{\\prime}}G^{d}_{\\sigma}(k)G^{d}_{\\bar{\\sigma}}(k+Q)\n \\Gamma^{\\sigma\\bar{\\sigma}}(Q)G^{d}_{\\sigma}(k^{\\prime})\n G^{d}_{\\bar{\\sigma}}(k^{\\prime}+Q)\n \\end{align}\n\\end{subequations}\nThe momentum sum over $\\vec{k}$ and $\\vec{k}^{\\prime}$ can be performed\nindependently by FFT becasue the DF vertx $\\Gamma^{\\sigma\\sigma^{\\prime}}(Q)$\nonly depends on the center of mass momentum $Q$.\n\nNow the z-component DF spin susceptibility $\\langle S^{z}\\cdot\nS^{z}\\rangle=\\frac{1}{2}(\\chi^{\\uparrow\\uparrow}_{d} \n-\\chi^{\\uparrow\\downarrow}_{d})$ can be determined from the spin channel\ncomponent calculated above. In Fig. \\ref{momentum-distribution}, \n$\\tilde{\\chi}^{zz}=\\chi^{zz}-\\chi_{0}^{zz}$ is shown for $U\/t=4$ at\ntemperatures $\\beta t = 4.0$ (left panel) and $\\beta t=1.0$ (right\npanel). With the lowing down of temperature the DF susceptibility grows up, \nespecially at wave vector $(\\pi, \\pi)$. The momentum $\\vec{k}_{x}$ and\n$\\vec{k}_{y}$ run from $0$ to $2\\pi$. \n \\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=230pt]{Sz}\n \\caption{The nontrivial part of the DF spin susceptibilities as a function\n of momentum in 2D Hubbard Model at $U\/t=4.0$, $\\beta t = 1.0$ (right\n panel) and $\\beta t = 4.0$ (left panel). Here 32 $\\times$ 32 momentum\n points are used in the first Brillouin zone.} \n \\label{momentum-distribution}\n \\end{center}\n\\end{figure}\nThe susceptibility is strongly peaked at the wave vector $(\\pi,\\pi)$ at the low\ntemperature case and the peak value becomes higher and higher. The magnetic\ninstability of the DF system is indicated by the enhancement of the\nDF susceptiblity. The effect of momentum dependence of vertex is clearly\nvisible in this diagram. The bare vertex which is only a function of frequency\nbecomes momentum dependent through the Bethe-Salpeter equation. Later on we\nwill see that such momentum dependent vertex plays a very important role in the\ncalculation of the lattice fermion susceptibility. \n \n\\section{Lattice susceptibility}\\label{application} \n\nThe strong antiferromagnetic fluctuation in 2D system is indicated by\nthe enhancement of the DF susceptibility at the wave vector\n$(\\pi,\\pi)$ shown in Fig \\ref{momentum-distribution}. This is the consequence\nof the deep relation between the the Green's function of the lattice and the\nDF, see Eq. (\\ref{relation}). In order to observe the magnetic\ninstability of the lattice fermion directly, we have calculated the\nlattice susceptibility based on the DF method. By differentiating\nthe partition function in Eqns. (\\ref{original_fermion},\n\\ref{auxiliary_field}) twice over the kinetic energy, we obtain an exact\nrelation between the susceptibility of DF and lattice fermions. After some\nsimplifications\\cite{Brener-2007}, it is given by \n\\begin{eqnarray}\n && \\chi_{f}(Q) = \\chi^{0}_{f}(Q) + \\nonumber\\\\\n && \\frac{T^{2}}{N^{2}}\\sum_{k,k^{\\prime}}G^{\\prime}(k)G^{\\prime}(k+Q)\n \\Gamma^{d}_{Q}(\\nu,\\nu^{\\prime})G^{\\prime}(k^{\\prime})G^{\\prime}\n (k^{\\prime}+Q)\n\\end{eqnarray} \nHere $G^{\\prime}$ cannt be interpreted as a particle propagator, it is\ndefined as: \n\\begin{equation}\n G^{\\prime}(k) = \\frac{G^{d}(k)}{g_{\\nu}[\\Delta_{\\nu}-\\epsilon(k)]}\n\\end{equation}\nAgain, the sum is performed over internal momentum and frequency $k,\nk^{\\prime}$ which is performed by FFT and rough summing over a few Matsubara\npoints. Again as in Eq. (\\ref{relation}), this equation established a\nconnection between the lattice susceptibility and the DF\nsusceptibility. From this point of view, it is easy to understand that the\ninstability of DFs generates the instability of the lattice\nfermions. \n\nOne can also find relations for the higher order Green's function of the\nDF and the lattice fermions in the same way. This emphasizes the similar\nnature of the DF and lattice fermions except that DF possess only\nnon-local information, since the DMFT self-consistency ensures that the local\nDF Green's function is exactly zero.\n\nThe lattice magnetic susceptibility is calculated using the following\ndefinition \n\\begin{eqnarray}\n \\chi_{m}(q) &=& \\frac{1}{N}\\sum_{i}e^{iq \\cdot\n r_{i}}\\int_{0}^{\\beta}d\\tau e^{-i\\omega_{m}\\tau}\\chi_{f}(i,\n \\tau)\\nonumber\\\\ \n &=& 2(\\chi_{f}^{\\uparrow\\uparrow}-\\chi_{f}^{\\uparrow\\downarrow})\n\\end{eqnarray} \nwhere $\\chi_{f}(i,\n\\tau)=\\langle\n[n_{i,\\uparrow}(\\tau)-n_{i,\\downarrow}(\\tau)]\\times[n_{0,\\uparrow}(0) -\nn_{0,\\downarrow}(0)]\\rangle$. $\\chi_{f}$ represents the lattice susceptibility\nin order to distinguish with that of the DF. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_0}\n \\caption{The uniform spin suscetibility of the DF using the bare\n vertex (only frequency dependent) and the full vertex(vertex from the\n Bethe-Salpeter quqation) for half filled 2D Hubbard model at $U\/t = 4.0$\n and various temperatures. These results reproduce the similiar solution\n in comparison with the calculation of finite size of\n QMC.}\\label{uniform_susceptibility} \n \\end{center}\n\\end{figure}\n\nWe have used two different ways to calculate the lattice susceptibility. First\nwe have solved the above equation using the bare vertex\n$\\Gamma(\\nu,\\nu^{\\prime}; \\omega)$ which is obtained from the DMFT \ncalculation. In contrast, the second calculation was performed using the full\nDF vertex. In both of these calculations, the full one particle DF Green's\nfunction was used. The momentum dependent of the DF vertex is\nobtained through the calculation of the Bethe-Salpeter equation. The lattice\nsusceptibility is expected to be improved if we use the momentum\ndependence DF vertex. In this way, we can understand the effect of momentum\ndependence in the DF vertex.\n\nIn Fig. \\ref{uniform_susceptibility} we plotted the results for the uniform\nsusceptibility $\\chi_{m=0}(0,0)$ by using both the bare and full DF\nvertex. The lattice QMC result\\cite{Moreo-1993} is shown for comparison. The\ncalculation is done for $U\/t = 4.0$ and several values of temperature. The\nmomentum sum is approximated over 32 $\\times$ 32 points here. Both of these\ncalculations reproduce the well known Curie-Weiss law behavior. Surprisingly\nenough, the results for the bare vertex fit the QMC results better than that\nfor the momentum dependent vertex. We believe that this is the finite size\neffect of QMC\\cite{Moreo-1993}. A. Moreo showed that $\\chi$ becomes smaller\nwhen increasing the cluster size $N$. The 4 $\\times$ 4 cluster calculation \nresult at the same temperature located above of that from 8 $\\times$ 8 cluster\ncalculation. Therefore the results obtained from the full vertex is expected\nto be more reliable. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_pi}\n \\caption{Uniform spin susceptibility at the wave vector $(\\pi, \\pi)$. The\n QMC results are obtained from Ref.\\cite{Bickers-1991(2)}.}\n \\label{chi_pi}\n \\end{center}\n\\end{figure}\n\nThe importance of the momentum dependence of the DF vertex is more clearly\nobserved in the calculation of $\\chi_{m}(\\pi, \\pi)$, see\nFig. \\ref{chi_pi}. Again, in this diagram QMC results\\cite{Bickers-1991(2)}\nare shown for comparison. The same parameters are used as in \nFig. \\ref{uniform_susceptibility}. The result from the DF with bare vertex\ndoes not produce the same results compared with QMC solution. Evenmore\ninteresting, with decreasing temperature the deviation becomes larger. On the\nother hand, the momentum dependent vertex in the DF method gives a\nsatisfactory answer. This shows the importance of the momentum\ndependence in the DF vertex function. Fig.~\\ref{chi_q} shows the evolution of\n$\\chi$ against $q$ for fixed transfer frequency $\\omega_{m}=0$. The path in\nmomentum space is shown in the inset. From this diagram we can see that\n$\\chi(q,0)$ reaches its maximum value at wave vector $(\\pi,\\pi)$. \n\nThe comparison between the DF and QMC results shows the good\nperformance of DF method. The DF calculation started from a\nsingle site DMFT calculation and by introducing an auxiliary field, the\nnon-local information is introduced and nicely reproduces the QMC results. Our\ncalculation could be done within four hours for each value of the\ntemperature on average. In this sense, this method is cheap and reliable\ncompared with the more computationally intensive lattice QMC calculation. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_q}\n \\caption{$\\chi(q,0)$ vs $q$ at $\\beta t= 2.0$, $U\/t = 4.0$ for various $q$\n which is along the trajectory shown in the inset.} \\label{chi_q} \n \\end{center}\n\\end{figure}\n\nSimilar as the DF method, Dynamical Vertex Approximation\n(D$\\Gamma$A)\\cite{Toschi-2007} is also based on the two particle local\nvertex. It deals with the lattice fermion directly, without introducing any\nauxiliary field. The perturbative nature of this method ensures its validity\nat weak-coupling regime. Unlike in the DF method, D$\\Gamma$A takes the\nirreducible two particle local vertex as building blocks. \n\\begin{subequations}\\label{DGA-BSE}\n \\begin{align}\n & \\gamma_{c(s)}^{-1}(\\nu,\\nu^{\\prime};\\omega) = \n \\gamma^{-1}_{c(s),ir}(\\nu,\\nu^{\\prime};\\omega) - \n \\chi_{0}(\\nu;\\omega)\\delta_{\\nu,\\nu^{\\prime}} \\\\\n & \\Gamma_{c(s)}^{-1}(\\nu,\\nu^{\\prime};Q) = \n \\gamma^{-1}_{c(s),ir}(\\nu,\\nu^{\\prime};\\omega) -\n \\chi_{0}(\\nu;Q)\\delta_{\\nu,\\nu^{\\prime}}\n \\end{align}\n\\end{subequations}\nwith the spin and charge vertex defined as\n$\\gamma_{c(s)}=\\gamma^{\\uparrow\\uparrow}\\pm\\gamma^{\\uparrow\\downarrow}$. The\nbare susceptibility is defined as \n\\begin{subequations}\n \\begin{align}\n & \\chi_{0}(\\nu;\\omega) = -TG_{loc}(\\nu)G_{loc}(\\nu+\\omega) \\\\\n & \\chi_{0}(\\nu,Q) = -\\frac{T}{N}\\sum_{\\vec{k}}\n G^{0}(\\vec{k},\\nu)G^{0}(\\vec{k}+\\vec{q},\\nu+\\omega)\n \\end{align}\n\\end{subequations}\nAnd the self-energy is calculated through the standard Schwinger-Dyson\nequation \n\\begin{equation}\\label{DGA-Selfenergy}\n \\Sigma(k) = -U\\frac{T^{2}}{N^{2}}\\sum_{k^{\\prime},Q}\n \\Gamma_{f}(k,k^{\\prime};Q)G^{0}(k^{\\prime})G^{0}(k^{\\prime}+Q)G^{0}(k+Q)\n\\end{equation}\nHere, the full vertex $\\Gamma_{f}(k,k^{\\prime};Q)$ is obtained by summing all\nthe channel dependent vertices and subtracting the double counted diagrams.\n\\begin{eqnarray}\\label{DGA-FullVertex}\n \\Gamma_{f}(k,k^{\\prime};Q)&=&\\frac{1}{2}\\bigg\\{\n [3\\Gamma_{c}(\\nu,\\nu^{\\prime};Q)-\\Gamma_{s}(\\nu,\\nu^{\\prime};Q)]\\nonumber\\\\\n &&-[\\Gamma_{c}(\\nu,\\nu^{\\prime};\\omega)-\n \\Gamma_{s}(\\nu,\\nu^{\\prime};\\omega)]\\bigg\\}\n\\end{eqnarray}\nThe one particle propagator is given by the DMFT lattice Green's function\nwhere the self energy is purely local $G^{0}(k)=\n1\/[i\\nu-\\epsilon(k)-\\Sigma(\\nu)]$, the local Green's function is $G_{loc}(\\nu)\n= 1\/[i\\nu-\\Delta(\\nu)-\\Sigma(\\nu)]$. Then the Dyson equation gives the lattice\nGreen's function from the self-energy function $G^{-1} = G^{-1}_{0}-\\Sigma$. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_0_DGA}\n \\caption{Comparison with the D$\\Gamma$A susceptibilities $\\chi(0,0)$ which\n obtained from both the DMFT lattice Green's function (D$\\Gamma$A\n $(G^{0}$)) and the full Green's function (D$\\Gamma$A $(G$)), see context\n for more details.}\n \\label{DGA_0}\n \\end{center}\n\\end{figure}\n\nBefore presenting the comparison, we take a deeper look at the analysis of \nEq. (\\ref{DGA-BSE}), \n\\begin{eqnarray}\n \\Gamma_{c(s)}^{-1}(\\nu,\\nu^{\\prime};Q) &=& \n \\gamma^{-1}_{c(s)}(\\nu,\\nu^{\\prime};\\omega) - \\nonumber \\\\\n &&[\\chi_{0}(\\nu;Q)-\\chi_{0}(\\nu,\\omega)]\\delta_{\\nu,\\nu^{\\prime}}\n\\end{eqnarray} \nThe second term in the brackets on RHS removes the local term from the\nbare susceptibility. The whole term in the brackets then represents only the\nnon-local bare susceptibility. In order to compare with the DF\nmethod, we take the inverse form of Eq. (\\ref{BSE})\n\\begin{eqnarray} \n \\Gamma_{d,c{s}}^{-1}(\\nu,\\nu^{\\prime};Q) &=&\n \\gamma_{c(s)}^{-1}(\\nu,\\nu^{\\prime}, \\omega) - \\nonumber\\\\\n && \\frac{T}{N}\\sum_{\\vec{k}}G^{d}(k)G^{d}(k+q) \n\\end{eqnarray}\nThe above two equations are same except for the last term. Since the local\nDF Green's function $G^{d}_{loc}$ is zero, the bare DF\nsusceptibility is purely non-local which coincides with the analysis of\nD$\\Gamma$A Bethe-Salpeter equation. Therefore, it is not surprising that \nthese two methods generate similar results. It is not easy to perform a term\nto term comparison between the DF method and D$\\Gamma$A although the\nbare susceptibilities have no local term in both of these method. The\none particle Green's functions have different meaning in these two methods. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=240pt]{chi_pi_DGA}\n \\caption{D$\\Gamma$A susceptibilities $\\chi(\\pi,\\pi)$ at $U\/t=4.0$. The\n susceptibility are determined from both of the DMFT and full lattice\n Green's function together with the vertex obtained from\n Eq. (\\ref{DGA-FullVertex})}. \n \\label{DGA_pi}\n \\end{center}\n\\end{figure}\n\nThe lattice susceptibility within the D$\\Gamma$A method is obtained by\nattaching four Green's functions on the vertex obtained in\nEq. (\\ref{DGA-FullVertex}). There are two possible choices of the lattice\nGreen's function, one is the DMFT lattice Green's function $G^{0}$, the other\none is the Green's function $G$ constructed by the non-local self-energy from\nthe Dyson equation. In Fig. \\ref{DGA_0} and \\ref{DGA_pi}, we presented\nthe D$\\Gamma$A lattice susceptibility calculated from both the DMFT lattice\nGreen's function labeled as D$\\Gamma$A($G^{0}$) and the full Green's function\nlabeled as D$\\Gamma$A($G$). The DF result from the calculation with\nthe full DF vertex is re-plotted for comparison. In Fig. \\ref{DGA_0}, the\nD$\\Gamma$A susceptibility calculated from the DMFT Green's function\n(D$\\Gamma$A($G^{0}$)) is basically the same as the DF susceptibility\nonly with some small deviation. The results for $T\/t > 1.0$ which are not shown\nhere which nicely repeat the DF and QMC results, the deviation\nbetween the D$\\Gamma$A and the DF method becomes smaller with the\nincreasing of temperature. The D$\\Gamma$A susceptibility is calculated from\nthe full Green's function (D$\\Gamma$A($G$)) shows a different behavior at low\ntemperature regime which reached its maximum value at $T\/t\\approx0.36$. As we\nknow, the Hubbard Model at half filling with strong coupling maps to the\nHeisenberg model, $\\chi$ reasches a maximum at $T\\approx J$ where \n$J$ is the effective spin coupling constant given as $4t^{2}\/U$. The\ncalculation uses the parameter $U\/t=4.0$ which is in the intermediate coupling\nregime. Therefore we further calculated the lattice susceptibility at\n$U\/t=10.0$ which are shown in Fig. \\ref{chi_0_U10}. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=220pt]{chi_0_U10}\n \\caption{The comparison of the DF resulsts and that of QMC for\n the uniform susceptibility at $U\/t=10$. 4$\\times$4 QMC\n results\\cite{Moreo-1993} also shows the errorbars.} \n \\label{chi_0_U10}\n \\end{center}\n\\end{figure}\n\nWhen the temperature is greater than 0.4, the DF method and\nD$\\Gamma$A (D$\\Gamma$A($G^{0}$)) generate the similar results to the QMC\ncalculation. Reducing the temperature further, the QMC susceptibility\ngreatly drops and shows a peak around 0.4 which coincides with the\nbehavior of the Heisenberg model. The DF femion and D$\\Gamma$A susceptibility\ncontinuously grows up with the decreasing of temperature. Although the\nD$\\Gamma$A with the full Green's function (D$\\Gamma$A($G$)) shows a peak, it\nlocates at $T\/t=0.6667$ which is larger than the peak position of the QMC. And\nD$\\Gamma$A($G$) generated a large deviation from that of QMC. In \nthis diagram, we only show the results of the DF approach for\n$T\/t>0.3$ and the D$\\Gamma$A results for $T\/t>0.4$. The Bethe-salpeter\nequation of the D$\\Gamma$A have a eigenvalue approaching one when further\nlowering the temperature, which makes the access of lower temperature region\nimpossible. \n \n\\begin{figure}[b]\n \\begin{center}\n \\includegraphics[width=220pt]{eigenvalue_T}\n \\caption{The evolution of maximum eigenvalue in spin channel against\n temperature for DF method and D$\\Gamma$A.}\n \\label{eigenvalue-T}\n \\end{center}\n\\end{figure}\n\nFig. \\ref{DGA_pi} shows the results of D$\\Gamma$A susceptibilities at wave\nvector $(\\pi, \\pi)$. In contrast to the comparison for $\\chi(0,0)$ results, the\nD$\\Gamma$A susceptibility calculated from the full Green's function D$\\Gamma$A\n($G$) yields better results than that from the calculation with the DMFT\nGreen's function D$\\Gamma$A ($G^{0}$). D$\\Gamma$A ($G$) results are almost on\ntop of the DF results, the results with DMFT Green's function D$\\Gamma$A\n($G^{0}$) is large than the DF results. The deviation becomes\nlarger at lower temperature. Summarizing, the D$\\Gamma$A calculation\nusing the full Green's function generated the same result as the DF\nmethod for $\\chi(\\pi,\\pi)$ while failed to produce $\\chi(0,0)$ correctly. In\ncontrast, the calculation with the DMFT Green's function in D$\\Gamma$A nicely\nproduced the results calculated with the DF method for $\\chi(0,0)$\nwhile generated larger devivation for $\\chi(\\pi,\\pi)$ at lower temperature\nregime compared to that from the DF method. Together with\nFig. \\ref{uniform_susceptibility} and \\ref{chi_pi}, we can see that the DF\nfermion calculation with the full DF vertex generated basically the same\nresults for both $\\chi(0,0)$ and $\\chi(\\pi,\\pi)$ compared to the results of\nQMC. \n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=230pt]{chi_0_Away}\n \\caption{Uniform magentic susceptibility is plotted as a function of\n dopping at $\\beta t=2.5$ and $U\/t =4.0, 10.0$.}\\label{away-half}\n \\end{center}\n\\end{figure}\n\nIn both the DF method and the D$\\Gamma$A, the operation of inverting\nlarge matrices is required for solving the Bethe-Salpeter\nequation. Fig. \\ref{eigenvalue-T} shows the leading eigenvalue of \nEqns. (\\ref{BSE}) and (\\ref{DGA-BSE}). As expected, the leading eigenvalue\napproaches one with decreasing temperature which directly indicates the\nmagnetic instability of 2D system. The eigenvalues corresponding to the DF\nfermion method always lies below of that from D$\\Gamma$A indicating the\nbetter convergence of the DF method. When the leading eigenvalues are closed to\none, the matrix inversion in Eqns. (\\ref{BSE}) and (\\ref{DGA-BSE}) are ill\ndefined, which prevents the investigation at very low temperature. \n\nConcerning the performance of the DF method, we also calculated the \nuniform susceptibility at away half-filling. In the strong-coupling limit,\nthe Hubbard model is equivalent to the Heisenberg model with coupling constant\n$J=4t^{2}\/U$. The consequence of doping is to effectively decrease the\ncoupling $J$, which yields the increasing behavior of $\\chi$ with doping. The\nfinite size QMC calulation\\cite{Moreo-1993, Chen-1993} observed a \nslightly increasing $\\chi$ with very small doping at strong interaction\nor in the low temperature region. Here, we did a similar calculation at $\\beta\nt=2.5$ and $U\/t=4, 10$. Since the DF method and the D$\\Gamma$A do not\nsuffer from the finite size problem. We would expect to observe results similar\nto those of QMC\\cite{Moreo-1993,Chen-1993}. In D$\\Gamma$A the\nsuseceptibility is calculated from the DMFT Green's function $G^{0}$ and the\nvertex obtained from Eq. (\\ref{DGA-FullVertex}). As shown in\nFig. \\ref{away-half} at $U\/t=4.0$, the susceptibility $\\chi$ slightly\nincreases in the weak dopping region where $\\delta$ is around $0.05$, DF\nfermion results clearly showed such behavior, D$\\Gamma$A also gave a signal of\nit. Further doping the system, both the D$\\Gamma$A and the DF method\nreproduce the decrease with doping as already seen in the QMC. With the\nincreasing of interaction, we would expect to see the enhancement of this\neffect, however our calculation indicates that such increasing-decreasing\nbehaviro dissappear. Both the D$\\Gamma$A and the DF method give the\nsame decreasing curve which contradict to QMC result\\cite{Moreo-1993}. The\nresults will most likely be further improved by including the higher order\nvertex or calculating the cluster DMFT plus DF\/D$\\Gamma$A\\cite{Hafermann-2007}.\n \n\\section{Conclusion}\\label{conclusion}\n\nIn this paper, we extended both the DF method and D$\\Gamma$A to\ncalculate the lattice susceptibility. Both of these methods gave equally good\nresults compared with QMC calculation at $U\/t=4.0$. Although they are supposed\nto be weak-coupling methods, at $U\/t=10.0$ these two methods generated right\nresults at high temperature region. While both of them failed to reproduce\nthe Heisenberg physics at low temperature. The investigation of the lattice\nsusceptibility suffers from hard determined matrix inversion problem at low\ntemperature regime. The DF methods always generates smaller eigenvalues\ncompared to D$\\Gamma$A indicating the better convergence. The implementation\nof DF method in momentum space greatly improves the calculational\nspeed and makes it easier to deal with larger size lattice. \n\n\\begin{acknowledgments}\nWe would like to thank the condensed matter group of\nA. Lichtenstein at Hamburg University for their hospitality\nin particular for the discussions and open exchange\nof data with H. Hafermann. Gang Li and Hunpyo Lee would like\nto thank Philipp Werner for his help in implementing the\nstrong-coupling CT-QMC code.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nReactions on surfaces play an important role in many technological \napplications like the heterogenous catalysis, growth of semiconductor \ndevices, corrosion and lubrication of mechanical parts, or hydrogen\nstorage in metals. In spite of this significance the understanding\nof the microscopic details of these reactions is still rather incomplete.\nOf particular importance are processes in which chemical bonds of\nmolecules are breaking due to the presence of a substrate because\nthere processes represent the first elementary step in, e.g.,\nheterogeneous catalysis or corrosion. Often this is the rate-limiting\nstep, for example in the ammonia synthesis. Hydrogen dissociation\non metal surfaces has become {\\em the} model system for the\nbond-breaking process on surfaces in the last years\nbecause it can be studied in detail experimentally as well as\ntheoretically. In particular in the theoretical description\nthere has been much progress recently due to the improvement of \ncomputer power and the development of efficient algorithms.\nIt has become possible to map out detailed potential energy surfaces\nof the dissociation of hydrogen on metal surfaces \nby density functional theory calculations \n\\cite{Ham94,Whi94,Wil95,Wil96PRB,Whi96PRB,Wie96,Eich96,Dong96}.\nThe availability of high-dimensional\nreliable potential energy surfaces has challenged the dynamics community\nto improve their methods in order to perform high-dimensional dynamical \nstudies on these potentials. Now quantum studies of the dissociation of \nhydrogen on surfaces are possible in which all six degrees of \nfreedom of the molecule are treated dynamically\n\\cite{Gro95PRL,Gro98PRB,Kro97PRL,Kro97JCP,Dai97}.\nIn this brief review I will illustrate this progress by focusing\non the hydrogen dissociation on the clean and sulfur-covered\npalladium surface. \n\nHydrogen is the simplest molecule which\nmakes it accessible to a relatively complete theoretical treatment.\nAt the same time hydrogen is also well-suited for performing\nexperiments which allows a fruitful interaction between theory\nand experiment. I will show that general concepts relating to\nthe reactivity of surfaces as well as to dynamical reaction\nmechanisms can be deduced from the detailed comparison\nof theoretical and experimental results of the hydrogen dissociation\nof metal surfaces. These concepts are applicable to any reaction\nsystem making hydrogen the ideal candidate\nfor studying reactions on surfaces.\n\n\n\n\n\\section{General concepts in the adsorption dynamics at surfaces}\n\nThe sticking or adsorption probability is defined as the fraction\nof atoms or molecules impinging on a surface that are not\nscattered back, i.e. that remain on the surface. It should be noted\nhere that there is no unambiguous definition of the sticking probability\nbecause for surfaces with non-zero temperature every adsorbed particle\nwill sooner or later desorb again. Hence the sticking probability\ndepends on the time-scale of the required residence time on the surface.\n\n\n\n\nAtomic adsorption is often very efficient. \nHydrogen atoms, e.g, stick at metal surfaces \\cite{Eilm96} and \nsemiconductor surfaces \\cite{Schu83} with a probability of order unity. \nHowever, dissociative adsorption probabilities can differ by many orders \nof magnitude. Whereas the sticking probability of hydrogen molecules on \nmany transition metal surfaces is about 0.5 \\cite{Eilm96,Ren89},\nat room temperature the dissociation probability of H$_2$\/Si\nis only 10$^{-8}$ \\cite{Bra96PRB}, and for N$_2$\/Ru it is even as low as\n10$^{-13}$ \\cite{Hin97}. The investigation of processes that occur within\nsuch a wide range of probabilities represents of course a great challenge\nfor the theory as well as for the experiment.\n\n\n\\begin{figure}[t]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n\\put(-1.,0.){ \\rotate[l]{\\epsfysize=10.cm \n \\epsffile{mol_ads_bw.ps}} }\n \\end{picture}\n\n\\end{center}\n \\caption{Atomic adsorption probability for Xe and Ar on Pt(111) and\n molecular adsorption probability of N$_2$\/W(100). These\n examples are taken from the textbook by Zangwill \n \\protect\\cite{Zan88}. The inset illustrates the adsorption\n process.\n } \n\n\\label{mol_ads}\n\\end{figure} \n\n\nThere is one fundamental difference between atomic and molecular adsorption \non the one side and dissociative adsorption on the other side that is very\nimportant for the theoretical description of these processes. Here with \nmolecular adsorption a sticking process is meant in which the molecule \nstays intact on the surface. This fundamental difference will be illustrated\nin the following. In atomic and molecular adsorption it is crucial that\nthe impinging particles transfer their kinetic energy to the surface,\notherwise they would be scattered back into the gas phase.\nIf $P_E (\\epsilon)$ is the probability that an incoming particle with\nkinetic energy $E$ will transfer the energy $\\epsilon$ to the surface,\nthen the atomic or molecular sticking probability can be expressed as\n\\begin{equation}\nS(E) \\ = \\ \\int_{E}^{\\infty} \\ P_E(\\epsilon) \\ d\\epsilon,\n\\end{equation}\ni.e., it corresponds to the fraction of particles that transfer more\nenergy to the surface than their initial kinetic energy. This excess\nenergy has to be transferred to substrate excitation, i.e., either\nphonons or electron-hole pairs. Hence any theoretical description\nof atomic or molecular adsorption has to consider dissipation to the continous\nexcitation spectrum of the substrate. In Fig.~\\ref{mol_ads}\nsticking probabilities for atomic and molecular adsorption as a function\nof the initial kinetic energy are shown that correspond indeed to textbook \nexamples \\cite{Zan88}. These curves show a typical behavior, namely the\ndecrease of the sticking probability with increasing kinetic energy.\nThis is due to the fact that the energy transfer to the surface becomes\nless efficient at higher kinetic energies. Of course, the higher the kinetic \nenergy is, the more energy is transfered to the surface. But the fraction\nof particles that loose more energy than their initial kinetic energy\nbecomes smaller at higher kinetic energy. There is still more\ninteresting physics in these sticking probabilities. For example, at low \nkinetic energies classically the sticking probability should become\nunity if there is no barrier before the adsorption well. Every\nimpinging particle transfers energy to the substrate so that\nin the limit of zero initial kinetic energy all particles will stick.\nQuantum mechanically, however, there is a non-zero probability for\nelastic scattering at the surface so that the sticking probabilities\nbecome less than unity in the zero-energy limit \\cite{Sch88}. In \nFig.~\\ref{mol_ads} these quantum effects at low energies are evident in \nthe sticking probability of the light noble gas argon on Pt(111) compared\nto the sticking probability of the heavier noble gas xenon on the same\nsurface.\n\n\n\\begin{figure}[t]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n\\put(-1.,0.){ \\rotate[l]{\\epsfysize=10.cm \n \\epsffile{diss_ads_bw.ps}} }\n \\end{picture}\n\n\\end{center}\n \\caption{Dissociative adsorption probability versus kinetic energy \n of H$_2$\/Cu(111) for molecules initially in the vibrational ground state. \n Solid line: results of five-dimensional calculations in which the molecular\n axis was kept parallel to the surface \n (from ref.~\\protect\\cite{Gro94PRL}).\n Dashed line: Experimental curve (from ref.~\\protect\\cite{Ret95}).\n The inset illustrates the dissociation process. }\n\n\\label{diss_ads}\n\\end{figure} \n\n\nNow in the case of dissociative adsorption there is another channel\nfor energy transfer, which is the conversion of the kinetic and internal \nenergy of the molecule into translational energy of the atomic fragments \non the surface relative to each other. This represents the fundamental\ndifference to atomic or molecular adsorption. It is true that eventually\nthe atomic fragments will also dissipate their kinetic energy and come\nto rest at the surface. However, especially in the case of light molecules\nlike hydrogen dissociating on metal surfaces the energy transfer to the\nsubstrate is very small due to the large mass mismatch. Whether a molecule\nsticks on the surface or not is almost entirely determined by the \nbond-breaking process for which the energy transfer to the substrate\ncan be neglected. This makes it possible to describe the dissociative\nadsorption process within low-dimensional potential energy surfaces\nneglecting the surface degrees of freedom\nif furthermore no substantial surface rearrangement upon adsorption occurs,\nas it is usually case in the dissociative adsorption on close-packed\nmetal surfaces. Fig.~\\ref{diss_ads} shows the dissociative adsorption \nprobability of a system which also corresponds to a textbook example,\nnamely the dissociative adsorption of H$_2$ on Cu(111). In this system\nthe dissociation is hindered by a noticeable barrier so that the\ndependence of the sticking probability on the kinetic energy exhibits\na behavior typical for activated systems \\cite{Gro94PRL,Ret95}.\n\n\n\n\\section{Dissociative adsorption at a transition metal surface}\n\nTransition metal surfaces are usually very reactive as fas as\nhydrogen dissociation is concerned \\cite{Ren89,Aln89,Ber92,But94}.\nIn Fig.~\\ref{h2pdstick} the results of molecular beam experiments\nof Rendulic, Anger and Winkler \\cite{Ren89} and of Rettner and\nAuerbach \\cite{Ret96} for the dissociative adsorption of H$_2$ \non Pd(100) are shown. At low kinetic energies\nthese experiments yield a sticking probability above 0.5. But even \nmore interestingly, the sticking probability initially decreases\nwith increasing kinetic energy. This is reminiscent of the dependence\nof the sticking probability on the kinetic energy in atomic or\nmolecular adsorption illustrated in Fig.~\\ref{mol_ads}. Therefore\nfor a long time it was believed that such an initially decreasing\nsticking probability in dissociative adsorption is a signature of\nthe so-called precursor mechanism \\cite{Ren94}. In this mechanism the \nmolecule does not directly dissociate but is first trapped molecularly\nin a precursor state from which it then dissociates. This trapping\nprobability decreases with increasing kinetic energy and thus\ndetermines the sticking at low kinetic energies.\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n\\centerline{ \\rotate[r]{\\epsfysize=8.cm \n \\epsffile{h2pd100.eps}} }\n\n \\end{picture}\n\\end{center}\n \\caption{Sticking probability versus kinetic energy for\na hydrogen beam under normal incidence on a Pd(100) surface.\nTheory: six-dimensional results for H$_2$ molecules initially in the \nrotational and vibrational ground state (dashed line)\nand with an initial rotational and energy distribution \nadequate for molecular beam experiments (solid line) \\protect\\cite{Gro95PRL}.\nH$_2$ molecular beam adsorption experiment under normal incidence\n(Rendulic {\\it et al.}~\\protect\\onlinecite{Ren89}): circles;\nH$_2$ effusive beam scattering experiment with an incident angle of\nof $\\theta_i = 15^{\\circ}$\n(Rettner and Auerbach~\\protect\\onlinecite{Ret96}): long-dashed line. }\n\\label{h2pdstick}\n\\end{figure} \n\n \n\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,11.0)\n\\put(-1.5,-0.5){ {\\epsfysize=12.cm \n \\epsffile{H2Pd_PES_inset_bw.eps}} }\n\\put(2.5,-0.5){ {\\epsfysize=12.cm \n \\epsffile{H2Pd_PES_hth_bw.eps}} }\n\n \\end{picture}\n\n\\end{center}\n \\caption{Contour plots of the PES along a two two-dimensional cuts \n through the six-dimensional coordinate space of \n H$_2$\/Pd\\,(100), so-called elbow plots, determined by GGA \n calculations \\protect\\cite{Wil95,Wil96PRB}. The coordinates \n in the figure are the H$_2$ center-of-mass distance \n from the surface $Z$ and the H-H interatomic distance $d$.\n The dissociation process in the $Zd$ plane is illustrated\n in the inset. The lateral H$_2$ center-of-mass coordinates \n in the surface unit cell and the orientation of the molecular \n axis, i.e. the coordinates $X$, $Y$, $\\theta$, and $\\phi$ \n are kept fixed for each 2D cut and depicted above the \n elbow plots. Energies are in eV per H$_2$ molecule.\n The contour spacing in a) is 0.1~eV, while it is 0.05~eV in b).}\n\n\\label{h2pdelbow}\n\\end{figure} \n\n\nWilke and Scheffler have performed density-functional theory (DFT) calculations\nof the interaction of H$_2$ with Pd(100) in order to elucidate the dissociation\nprocess. In Fig.~\\ref{h2pdelbow} two so-called elbow plots are shown.\nThey represent a two-dimensional cut through the potential energy surface (PES)\nof H$_2$\/Pd(100) in which the orientation of the molecule, its \ncenter-of-mass lateral coordinates and the substrate are kept fixed. \nThe molecule is oriented parallel to the surface, and the PES is plotted\nas a function of the center-of-mass distance of the molecule from the\nsurface $Z$ and the interatomic distance $d_{\\rm H-H}$. Fig.~\\ref{h2pdelbow}a \ndemonstrates that the dissociation of H$_2$ on Pd(100) is non-activated, i.e.,\nthere are reaction paths towards dissociative adsorption with no energy\nbarrier. The majority of reaction pathways, however, is hindered by\nbarriers \\cite{Gro96CPLa}. \nFurthermore, in these calculations no molecular adsorption\nstate has been found. It looks like there is such a well in \nFig.~\\ref{h2pdelbow}b. However, the detailed DFT study of the PES\nhas shown that this apparent well does not correspond to a local minimum\nof the PES, it is rather a saddle point in the multi-dimensional PES.\n\n\nAn analytical representation of this {\\it ab initio} PES has been used\nfor a quantum dynamical study in which all six hydrogen degrees of freedom \nwere taken into account explicitly while the substrate was kept fixed\n\\cite{Gro96CPLa}. The results are also plotted in Fig.~\\ref{h2pdstick}.\nThe sticking curve for a monoenergetic beam initially in the vibrational and\nrotational ground state shows a strong oscillatory structure which will be \ndiscussed below. An experimental molecular beam, however, does not correspond\nto a monoenergetic beam in one specific quantum state. \nIf one assumes an energy spread and a distribution of internal\nmolecular states typical for a beam experiment, the oscillations\nare almost entirely smoothed out in the 6D quantum results\n(solid line in fig.~\\ref{h2pdstick}). The results corresponding to the\nbeam simulation agree with the experimental results semi-quantitatively.\nMore importantly, they reproduce the general trend found in the \nexperiment, namely the initial decrease of the sticking probability\nas a function of the kinetic energy followed by an increase at higher\nenergies. Now in the {\\it ab initio} PES there is no molecular adsorption\nstate, furthermore in the 6D quantum dynamical calculation no energy\ntransfer to the substrate is considered. Hence the precursor mechanism\ncannot be operative in the simulation. So what is the reason for the\ninitial decrease of the sticking probability?\n\nSince energy transfer cannot play a crucial role in the adsorption process,\nthis decrease in the sticking probability has to be caused by a purely \ndynamical effect, namely the steering\neffect \\cite{Gro95PRL,Gro95JCP,King78,Kay95,Whi96}:\nAlthough the majority of pathways to dissociative adsorption\nhas non-vanishing barriers with a rather broad distribution of\nheights and positions, slow molecules can be very efficiently\nsteered to non-activated pathways towards dissociative adsorption \nby the attractive forces of the potential. This mechanism becomes \nless effective at higher kinetic energies where the molecules are \ntoo fast to be focused into favourable configurations towards \ndissociative adsorption. If the\nkinetic energy is further increased, the molecules will eventually\nhave enough energy to directly traverse the barrier region leading\nto the final rise in the sticking probability.\n\n\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n \\special{psfile=h2pd_snap.ps \n vscale=85 hscale=85 angle=-90 voffset=370 hoffset=-240 }\n\n \\end{picture}\n\n\\end{center}\n \\caption{Snapshots of classical trajectories of hydrogen molecules \nimpinging on a Pd(100) surface. The initial conditions are chosen in \nsuch a way that the trajectories are restricted to the $xz$-plane.\nLeft trajectory: initial kinetic energy $E_i = 0.01$~eV. \nRight trajectory: same initial conditions as in the left trajectory\nexcept that the molecule has a higher kinetic energy of 0.12 eV. }\n\n\\label{traj2run}\n\\end{figure} \n\n\n\nIn order to illustrate the steering effects, we use the results\nof classical molecular dynamics calculations which have been\nperformed on exactly the same PES as the quantum dynamical calculations\n\\cite{Gro97Vac}. In these classical calculations significant\ndifferences in the sticking probability compared to the quantum\nresults have been found which are mainly due to zero-point effects.\nThe steering effect, however, is a general mechanism operative in\nquantum as well as in classical dynamics. I will therefore use\nsnapshots of two typical trajectories in order to illustrate\nthe dynamical mechanism responsible for the initial decrease\nof the sticking probaiblity. These trajectories are plotted\nin Fig.~\\ref{traj2run}. The initial conditions are chosen in \nsuch a way that the trajectories are restricted to the $xz$-plane.\nThe left trajectory demonstrates why the sticking probability\nis so large at low kinetic energies due to the steering effect.\nThe incident kinetic energy is $E_i = 0.01$~eV. In this particular \nexample the molecular axis is initially almost perpendicular to the\nsurface. In such a configuration the molecule cannot dissociate\nat the surface. But the molecule is so slow that the forces \nacting upon it can reorient the molecule. It is turned parallel \nto the surface and then follows a non-activated path towards \ndissociative adsorption. This shows how molecule with unfavorable\ninitial conditions can still dissociate due to very efficient\nsteering towards favorable configurations.\n\nThis process becomes less effective at higher kinetic energies,\nwhich is demonstrated with the right trajectory in Fig.~\\ref{traj2run}.\nThe initial conditions are the same as for the left trajectory, except \nfor the higher kinetic energy of 0.12~eV. Of course the same forces\nact upon the molecule, and due to the anisotropy of the PES the molecule also\nstarts to rotate to a configuration parallel to the surface. However,\nnow the molecule is too fast to finish this rotation. It hits the \nrepulsive wall of the PES at the surface with its molecular axis tilted by \nabout 45$^{\\circ}$ with respect to the surface normal. At the classical \nturning point there is a very rapid rotation corresponding\nto a flip-flop motion, and then the molecule is scattered back into \nthe gas-phase rotationally excited.\n\n\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.1)\n \\special{psfile=h2pd_scat.ps \n vscale=35 hscale=35 angle=-90 voffset=200 hoffset=-15 }\n \\end{picture}\n\n\\end{center}\n \\caption{Angular distribution of the in-plane and out-of-plane\n of scattering of H$_2$\/Pd(100). \n The initial kinetic energy is $E_i = 76$~meV, the\n incident angle is $\\theta_i = 32^{\\circ}$ along the\n $\\langle0 \\bar 1 1\\rangle$ direction. The molecules are initially\n in the rotational ground state $j_i = 0$.\n Open circles correspond to rotationally elastic, filled\n circles to rotationally inelastic diffraction.\n The radii of the circles are proportional to the logarithm \n of the scattering intensity. \n $x$ denotes the $\\langle0 \\bar 1 1\\rangle$ direction, $y$ the\n $\\langle0 1 1\\rangle$ direction. The specular peak is the largest\n open circle (from Ref.~\\protect\\cite{Gro96CPLb}). }\n\n\\label{inout}\n\\end{figure} \n\nNow I like to come back to the strong oscillatory structure of the\nsticking curve for a monoenergetic beam initially in one particular\nquantum state in Fig.~\\ref{h2pdstick}. \nThese oscillations are a consequence of the quantum\nnature of the hydrogen particle, in classical calculations they\ndo not appear \\cite{Gro98PRB,Gro97Vac}. If a quantum particle is\ninteracting with a periodic surface, coherent scattering leads to\ndiffraction. Such a calculated diffraction pattern is shown in \nFig.~\\ref{inout} for hydrogen molecules in the ground state\nscattering at a Pd(100) surface with an kinetic energy of $E_i = 76$~meV under \nan angle of incidence of $\\theta_i = 32^{\\circ}$. There are rather\nmany diffraction peaks since also rotationally inelastic diffraction \ncan occur, i.e., scattering in which the rotational state of the\nmolecule is changed. Still the number of diffraction\npeaks is finite and increases discontinously with increasing energy.\nAn analysis of the oscillatory structure of the sticking\nprobability in Fig.~\\ref{h2pdstick} reveals that these oscillations\ncan be related to threshold effects associated with the opening of \nnew scattering channels \\cite{Gro96CPLb,Gro96PRL}.\nThese oscillations have not been observed experimentally yet,\nalthough they have been carefully searched for \\cite{Ret96,Ret96PRL}.\nThey are very sensitive to the symmetry of the initial conditions\n\\cite{Gro96CPLb,Gro96PRL}. Furthermore, since Pd(100) is a very\nreactive surface, a large fraction of the incoming hydrogen molecules\nis not scattered back coherently but adsorbs dissociatively. These adatoms\nthen disturb the periodicity of the surface and thus suppress,\nin addition to already existing surface imperfections like steps\nand vacancies, the coherence of the scattering events. Hence \nthe experimental observation of this oscillations actually represents\na challenging task. \n\n\nThe dependence of adsorption and desorption \non kinetic energy, molecular rotation and orientation\n\\cite{Gro95PRL,Gro96SSb}, molecular vibration \\cite{Gro96CPLa}, ro-vibrational\ncoupling \\cite{Gro96Prog}, angle of incidence \\cite{Gro98PRB}, \nand the rotationally elastic and inelastic \ndiffraction of H$_2$\/Pd(100) \\cite{Gro96CPLb} have been studied so far\nby six-dimensional {\\it ab initio} dynamics calculations\non the same PES. The results of these calculations have been compared \nto a number of independent experiments \\cite{Ren89,Sch92,Beu95,Wet96}, and \nthey are at least in semi-quantitative agreement with all of these experiments\nThis shows that {\\it ab initio} dynamics calculations are indeed\ncapable of adequately describing the hydrogen dissociation\non transition metal surfaces. \n\n\n\n\\section{Dissociative adsorption at a sulfur-covered transition metal surface}\n\n\n\nThe presence of an adsorbate on a surface can profoundly change the \nsurface reactivity. A well-known example is the reduction of the activity\nof the car-exhaust catalyst by lead. But also adsorbed sulfur ``poisons''\nthis catalyst. An understanding of the underlying mechanisms and their \nconsequences on the reaction rates is therefore of decisive importance\nfor, e.g., designing better catalysts. \nTraditionally an ``trial and error'' approach\nwas used to improve the activity of a catalyst by adding some\nsubstances. \nOn Pd(100) it is experimentally well-known that the presence of sulfur \nleads to a large reducting of hydrogen dissociation \nprobability \\cite{Ren89,Bur90}. While at the clean surface the dissociation\nprobability is about 60\\% for a kinetic energy of $E_{\\rm i} = 0.05$~eV,\nat the sulfur-covered surface it drops below 1\\% at the same\nenergy \\cite{Ren89}.\n\nTheoretically this problem had only been addressed by a small \nnumber of studies. These focused either on the \nadsorbate induced change of the density of states (DOS) at the Fermi level\n\\cite{Fei84,Fei85,MacL86} or on the adlayer induced electrostatic\nfield \\cite{Nor84,Nor93,Ham93}. Just recently, the poisoning of hydrogen \ndissociation on Pd(100) by sulfur adsorption has been the subject of\ndetailed DFT studies \\cite{Wil95,Wil96S,Wei97}.\n\n\n\n\\begin{figure*}[t]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(18,8.5)\n\\put(-1.,-4.){ \\rotate[r]{\\epsfysize=20.cm \n \\epsffile{H2SPd_PES_applphys.ps}} }\n \\end{picture}\n\n\\end{center}\n\\caption{ Cuts through the six-dimensional potential \n energy surface (PES) of H$_2$ dissociation over\n (2$\\times$2)S\/Pd(100) at four different sites with\n the molecular axis parallel to the surface:\n a) at the fourfold hollow site;\n b) at the bridge site between two Pd atoms;\n c) on top of a Pd atom;\n d) on top of a S atom.\n The energy contours, given in eV per molecule, \n are displayed as a function of the H-H distance, \n $d_{H-H}$, and the height $Z$\n of the center-of-mass of H$_2$ above the topmost Pd layer. \n The geometry of each dissociation pathway is indicated \n in the panel above the contour plots. The large open circles\n are the sulfur atoms, the large filled circles are the\n palladium atoms.}\n\n\\label{H2SPd_PES}\n\\end{figure*} \n\n\n\n\n\nIn Fig.~\\ref{H2SPd_PES} I have collected four elbow plots of the hydrogen\ndissociation on the (2$\\times$2) sulfur covered Pd(100) surface \\cite{Wei97}.\nIn contrast to the clean Pd(100) surface, the hydrogen dissociation \non the sulfur covered surface is no longer non-activated. The minimum\nbarrier, which is shown in Fig.~\\ref{H2SPd_PES}a, has a height of\n0.1~eV and corresponds to a configuration in which the H$_2$ center of\nmass is located above the fourfold hollow site. This is the\nsite which is farthest away from the sulfur atoms in the surface unit cell.\nRecall that the most favorable reaction path on the {\\em clean} Pd(100)\nsurface corresponds to the H$_2$ molecule dissociating at the bridge position\nbetween two Pd atoms (see Fig.~\\ref{h2pdelbow}a). For this approach\ngeometry the dissociation at the sulfur covered surface is now\nhindered by a barrier of height 0.15~eV (Fig.~\\ref{H2SPd_PES}b). \nThere is a peculiar local minimum at the dissociation path for this\nconfiguration when the molecule is still 1\\mbox{\\AA} above the Pd atoms.\nThere are apparently subtle compensating effects between the attractive\ninteraction of H$_2$ with the Pd atoms and the repulsion originating\nfrom the S atoms. Figs.~\\ref{H2SPd_PES}a and b show \nthat the dissociation is hindered by the formation of energy barriers\nin the entrance channel of the potential energy surface, however, \nthe hydrogen dissociation is still exothermic, i.e., the poisoning\nis not due to site-blocking. This result is actually at variance\nwith measurements of the hydrogen saturation coverage as a function\nof the sulfur coverage \\cite{Bur90}.\nIn these experiments a linear decrease of the hydrogen saturation \ncoverage with increasing sulfur coverages was found. At a sulfur coverage\nof $\\Theta_{\\rm S} = 0.28$, which is close to the one used in the\ncalculations, hydrogen adsorption should be completely suppressed, i.e.,\nthere should be no attractive sites for hydrogen adsorption any more.\nA possible explanation for this apparent contradiction will be given\nbelow.\n\nCloser to the sulfur atoms the PES becomes strongly repulsive.\nThis is illustrated in Fig.~\\ref{H2SPd_PES}c and d. While the dissociation \npath over the Pd on-top position on the clean surface is hindered by a \nbarrier of height 0.15~eV \\cite{Wil96PRB} (Fig.~\\ref{h2pdelbow}b), \nthe adsorbed sulfur leads to an increase in this barrier height \nto 1.3~eV (Fig.~\\ref{H2SPd_PES}c). Directly above the sulfur atoms \nthe barrier towards dissociation even increases to values \nof about 2.5~eV for molecules oriented parallel to\nthe surface (Fig.~\\ref{H2SPd_PES}d). \n\n\nThe goal of any theoretical study should be to provide a qualitative\npicture that explains the calculated results. There are many different\nways of illustrating the electronic factors that determine the\nreactivity of a particular system \n(see, e.g., Refs.~\\cite{Wil96PRB,Eich96,Fei84,Ham95PRL,Wil96AP}). \nCurrent studies have emphasized that the reactivity of surfaces\ncannot be solely understood by the electronic density of states\nat the Fermi level \\cite{Ham95,WCoh96}. In order to understand the origins of \nthe formation of the small energy barriers at the hollow and bridge site and \nthe large energy barriers at the top sites, we will therefore compare the \nwhole relevant DOS for the H$_2$ molecule in these different geometries. \nFor a discussion of the reactivity of the clean Pd(100) surface I refer\nto Ref.~\\cite{Wil96PRB}. Here I focus on the changes of the density of states \ninduced by the presence of sulfur on the surface (Fig.~\\ref{DOS}).\n\n\n\n\\begin{figure*}[t]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(18,6.5)\n\\put(-3.,-8.){ \\rotate[r]{\\epsfysize=24.cm \n \\epsffile{H2SPd_DOS.ps}} }\n \\end{picture}\n\n\\end{center}\n \\caption{ Density of states (DOS) for a H$_2$ molecule situated at \n (a) ($ Z,d_{\\rm H-H}$)= (4.03\\AA, 0.75\\AA) and\n (b) ($Z,d_{\\rm H-H}$)= (1.61\\AA, 0.75\\AA) above the fourfold\n hollow site which corresponds to the configuration\n depicted in Fig.~\\protect\\ref{H2SPd_PES}a,\n and for a H$_2$ molecule situated at\n (c) (Z,d$_{H-H}$)= (3.38\\AA, 0.75\\AA) above the sulfur\n atom which corresponds to the configuration\n depicted in Fig.~\\protect\\ref{H2SPd_PES}d.\n $Z$ and $d_{\\rm H-H}$ denote the H$_2$ center-of-mass distance \n from the surface and the H-H interatomic distance, respectively.\n Given is the local DOS at the H atoms, the S adatoms, \n the surface Pd atoms, and the bulk Pd atoms. \n The energies are given in eV.}\n\n\\label{DOS}\n\\end{figure*} \n\n\nThe information provided by the density of states alone is often\nnot sufficient to assess the reactivity of a particular system.\nIt is also important to know the character of the occupied and\nunoccupied states. For the dissociation the occupation\nof the bonding $\\sigma_{\\rm g}$ and the anti-bonding $\\sigma_{\\rm u}^*$ \nH$_2$ molecular levels {\\em and} of the bonding and anti-bonding\nstates with respect to the surface-molecule interaction are of\nparticular importance.\n\n\n\nFigure~\\ref{DOS}a shows the DOS when the H$_2$ molecule is \nstill far away from the surface above the fourfold hollow site,\ni.e, in the configuration that corresponds to Fig.~\\ref{H2SPd_PES}a.\nThe H-H distance $d$ is 0.75 \\AA\\ and the center of mass of the H$_2$ \nmolecule is 4.03 \\AA\\ above the topmost Pd layer so that \nthere is no interaction between the hydrogen molecule and \nthe sulfur covered palladium surface.\nThe large peak in the sulfur DOS at -13~eV corresponds to the S 3$s$ state.\nThe sulfur {\\it p} orbitals strongly interact with the Pd {\\it d} states, \nwhich is evident from the peak in the sulfur DOS at the \nPd {\\it d} band edge (at E-E$_F$ = -4.8 eV) and from\nthe broad band at higher energies which has substantial \nweight close to the Fermi level. \nThe {\\it d} band at the surface Pd atoms is broadened and shifted\ndown somewhat with respect to the clean surface due to the interaction \nwith the S atoms \\cite{Wil96S}. There is one intense peak in the\nhydrogen DOS at -4.8 eV which corresponds to the $\\sigma_{\\rm g}$ state.\nThis peak is degenerate with the sulfur related bonding state at -4.8 eV, \nthis degeneracy, however, is accidental, as will become evident \nimmediately.\n\n\nThe density of states for the molecule at the minimum barrier position\nof Fig.~\\ref{H2SPd_PES}a is shown in Fig.~\\ref{DOS}b. Now the\n$\\sigma_{\\rm g}$ state has shifted down to -7.1 eV while the\nsulfur state at -4.8 eV remains almost unchanged. This indicates that\nthere is no direct interaction between hydrogen and sulfur. It also\nproofs that the degeneracy between these two states in Fig.~\\ref{DOS}a\nis accidental. Furthermore, we find a broad distribution of hydrogen \nstates with a small, but still significant weight below the Fermi level.\nThese are states of mainly H$_2$-surface antibonding character\n\\cite{Wil96S,Wei97}. A comparison with the hydrogen dissociation\nat the clean Pd surface yields that more H$_2$-surface antibonding\nstates are populated at the sulfur covered surface. This is caused\nby the sulfur induced downshift of the Pd $d$-band. These \nH$_2$-surface antibonding states lead to a repulsive interaction and thus \nto the building up of the barriers in the entrance channel of\nthe PES \\cite{Wil96S}. It is therefore an indirect interaction\nbetween sulfur and hydrogen that is responsible for the barriers\nat this site. A similar picture explains why for example noble\nmetals are so unreactive for hydrogen dissociation: The low-lying\n$d$-bands of the noble metals cause a downshift and a substantial \noccupation of the antibonding H$_2$-surface states resulting\nin high barriers for hydrogen dissociation \\cite{Ham95PRL}. \n\n\nThe situation is entirely different if the molecule approaches\nthe surface above the sulfur atom. This is demonstrated\nin Fig.~\\ref{DOS}c. The center of mass of the H$_2$ molecule is\nstill 3.38~\\AA\\ above the topmost Pd layer, but already at this\ndistance the hydrogen and the sulfur states strongly couple. The intense \npeak of the DOS at \\mbox{-4.8 eV} has split into a sharp bonding state \nat -6.6 eV and a narrow anti-bonding state at -4.0 eV.\nThus it is a direct interaction of the hydrogen with the sulfur related\nstates that causes the high barriers towards hydrogen dissociation\nclose to the sulfur atoms. In conclusion, the poisoning of hydrogen\ndissociation on Pd(100) by adsorbed sulfur is due to a combination\nof a indirect effect, namely the sulfur-related downshift of the Pd $d$-bands \nresulting in a larger occupation of H$_2$-surface antibonding states,\nwith a direct repulsive interaction between H$_2$ and S close to the\nsulfur atoms. \n\nIn order to assess the dynamical consequences of the sulfur adsorption\non the hydrogen dissociation, six-dimensional dynamical calculations \non the analytical representation of the {\\it ab initio} PES of H$_2$ at \nS(2$\\times$2)\/Pd(100) have been performed \\cite{Gro98PRL}.\nThe results of these quantum and classical calculations for the\nH$_2$ dissociative adsorption probability as a function of the incident\nenergy are compared with the experiment \\cite{Ren89} in Fig.~\\ref{stick}.\nIn addition, also the integrated barrier distribution $P_b(E)$,\n\\begin{eqnarray} \\label{barr}\nP_b (E) & = & \\frac{1}{2\\pi A} \\ \\int \n\\Theta (E - E_{\\rm b} (\\theta, \\phi, X, Y)) \\nonumber \\\\\n& & \\times \\ \\cos \\theta d\\theta \\ d\\phi \\ dX \\ dY\n\\end{eqnarray}\nis plotted. Here $\\theta$ and $\\phi$ are the polar and azimuthal\norientation of the molecule, $X$ and $Y$ are the lateral coordinates of \nthe hydrogen center-of-mass. $A$ is the area of the surface unit cell.\nEach quadruple defines a cut through the six-dimensional space (see\nFig.~\\ref{H2SPd_PES} for examples), \nand $E_{\\rm b}$ is the minimum energy barrier \nalong such a cut. The function $\\Theta$ is the Heavyside step function. \nThe quantity $P_b(E)$ is the fraction \nof the configuration space for which the barrier towards dissociation\nis less than $E$. If there were no steering effects, $P_b(E)$ would\ngive the classical sticking probability, i.e., it corresponds to the \nsticking probability in the classical sudden approximation or the so-called \n``hole model'' \\cite{Kar87}. \n\n\n\n\n\\begin{figure}[tb]\n\\unitlength1cm\n\\begin{center}\n \\begin{picture}(10,6.5)\n\\centerline{ \\rotate[r]{\\epsfysize=8.cm \n \\epsffile{h2spd_stick.eps}} }\n \\end{picture}\n\n\\end{center}\n \\caption{Sticking probability versus kinetic energy for\na H$_2$ beam under normal incidence on a S(2$\\times$2)\/Pd(100) surface.\nFull dots: experiment (from ref.~\\protect\\cite{Ren89});\nDashed-dotted line: Integrated barrier distribution,\nwhich corresponds to the sticking probability in the hole \nmodel \\protect\\cite{Kar87};\nSolid line: Quantum mechanical results for molecules initially in the\nrotational and vibrational ground-state;\nDashed line: Classical results for initially non-rotating and non-vibrating\nmolecules. The inset shows the quantum and classical results at low \nenergies.}\n\n\\label{stick}\n\\end{figure} \n\n\nFirst of all it is evident that the calculated sticking probabilities are \nsignificantly larger than the experimental results. \nOnly the onset of dissociative adsorption at $E_{\\rm i} \\approx 0.12$~eV \nis reproduced by the calculations. This onset is indeed also in agreement\nwith the experimentally measured mean kinetic energy of hydrogen \nmolecules desorbing from sulfur covered Pd(100) \\cite{Com80}.\nThe question arises where these large differences between theory and\nexperiment come from. It might be that uncertainties in the experimental \ndetermination are responsible for the difference. \nThe exact sulfur coverage in the \nexperiment was not very well characterized. The sulfur adlayer was obtained \nby simply heating up the sample which leads to segregation of bulk sulfur \nat the surface. The sulfur coverage was then monitored\nthrough the ratio of the Auger peaks S$_{132}$\/Pd$_{330}$ \\cite{Ren89}.\nSince the hydrogen sticking probability depends sensitively on the\nsulfur coverage \\cite{Ren89,Bur90}, a small uncertainty in the\nsulfur coverage can have a decisive influence. However, as noted above,\nwhile the DFT calculations yield that the poisoning is caused by\nthe building up of barriers hindering the dissociation, the vanishing\nhydrogen saturation coverage for roughly a quarter monolayer of adsorbed\nsulfur \\cite{Bur90} suggests that any attractive adsorption sites \nfor hydrogen have disappeared due to the presence of sulfur.\nThese seemingly contradicting\nresults and also the discrepancy between calculated and measured molecular\nbeam sticking probabilities could be reconciled if subsurface sulfur \nplays an important role for the hydrogen adsorption energies.\nSubsurface sulfur is not considered in the calculations but \nmight well be present in the experimental samples. The possible\ninfluence of subsurface species on reactions at surfaces certainly\nrepresents a very interesting and important research subject\nfor future investigations.\n\nExcept for this open question, there are further interesting\nresults obtained by the dynamical calculations. The calculated\nsticking probabilities are not only much larger than the\nexperimental ones, they are also much larger than what one would\nexpect from the hole model. This demonstrates that steering is\nnot only operative for potential energy surfaces with non-activated\nreaction paths like for H$_2$\/Pd(100), but also for activated\nsystems as H$_2$\/S(2$\\times$2)\/Pd(100). As Fig.~\\ref{H2SPd_PES} \ndemonstrates, the sulfur covered Pd surface represents a \nstrongly corrugated system with barrier heights varying by\nmore than 2~eV for molecules with their axis parallel to the\nsurface. And the large barriers above the sulfur atoms extend\nrather far into the gas phase (see Fig.~\\ref{H2SPd_PES}d).\nMolecules with unfavorable initial conditions are very effectively\nreoriented to low-barrier sites \\cite{Gro98PRL}. This leads to\nan enhancement of the sticking probability with respect to the\nhole model by a factor of three to four. \n\n\nFigure \\ref{stick} shows furthermore that the classical molecular\ndynamics calculations over-estimate the sticking probability \nof H$_2$ at S(2$\\times$2)\/Pd(100) compared to the quantum results. \nAt small energies below the minimum barrier height the quantum calculations \nstill show some dissociation due to tunneling, as the inset of \nFig.~\\ref{stick} reveals, whereas the classical results are of course zero. \nBut for higher energies the classical sticking probability is up to \nalmost 50\\% larger than the quantum sticking probabilities. \nThis suppression is also caused by the large corrugation and the\nanisotropy of the PES. The wave function describing the \nmolecule has to pass narrow valleys in the PES in the angular and lateral \ndegrees of freedom in order to dissociate. This leads to a localization\nof the wave function and thereby to the building up of zero-point energies\nwhich act as additional effective barriers. While\nthe vibrational H-H mode becomes softer upon dissociation so that\nthe zero-point energy in this particular mode decreases,\nfor the system H$_2$\/S(2$\\times$2)\/Pd(100) this\ndecrease is over-compensated by the increase in the zero-point\nenergies of the four other modes perpendicular to the reaction\npath, i.e., the sum of {\\em all} zero-point energies increases\nupon adsorption \\cite{Gro98PRL}. Therefore the quantum particles\nexperience an effectively higher barrier region causing the\nsuppressed sticking probability compared to the classical\nparticles. Interestingly enough, if the sum of all zero-point energies\nremains approximately constant along the reaction path as in the\nsystem H$_2$\/Pd(100), then these quantum effects almost cancel \nout \\cite{Gro98PRB,Gro97Vac}. \n\n\n\n\\section{Conclusions}\n\nIn this review {\\it ab initio} studies of reactions\non surfaces have been presented. In the last years the interaction between\nelectronic structure calculations on the one side and dynamical calculations\non the other side has been very fruitful. The availability of high-dimensional\nreliable potential energy surfaces has challenged the dynamics community\nto improve their methods in order to perform high-dimensional dynamical \nstudies on these potentials. Now quantum studies of the dissociation of \nhydrogen on surfaces are possible in which all six degrees of \nfreedom of the molecule are treated dynamically. In this review I have\ntried to show that this achievement represents an important step forward \nin our understanding of the interaction of molecules with surfaces.\nNot only the quantitative agreement with experiment is improved, but\nalso important qualitative concepts emerge from the electronic structure\ncalculations as well as from the high-dimensional dynamical simulations.\nThese concepts are applicable to any reaction system. This represents the \nimportance of hydrogen as a model system for studying reactions on surfaces.\nHowever, the {\\it ab initio} treatment of reactions on surfaces has now \nmatured enough so that in the future there will be also more studies\non other reaction systems like for example the important class of\noxidation reactions on surfaces. This will be the next step towards\na full microscopic description of catalytic reactions, one of the\nultimate goals of surface science. \n\n\n\\section*{Acknowledgements}\n\nI am very grateful to my collegues and coworkers who have made this work \npossible. I would like to mention in particular Thomas Brunner, \nBj{\\o}rk~Hammer, Ralf~Russ, Ching-Ming Wei, Steffen~Wilke, \nand last but not least my supervisors\nHelmar Teichler, Wilhelm Brenig and Matthias Scheffler.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Verification approaches}\nIn this section, we briefly discuss our experience in using hybrid system verification tools. \n{\\bf SpaceEx\\\/}~\\cite{DBLP:conf\/cav\/FrehseGDCRLRGDM11}, is a well-established reachability analysis tool for linear and affine hybrid systems. It implements the support function-based reachability algorithm, includes the PHAVer algorithm for rectangular dynamics~\\cite{Frehse:hscc05}, and also a simulator for nonlinear models. \nThe support function representation of sets is amenable to effective computation of convex hulls, linear transforms, Minkowski sums, etc.---operations that are necessary for safety verification. \n\n{\\bf C2E2\\\/}~\\cite{DuggiralaMVP15,FanQM0D16:CAV} is a simulation-driven bounded verification tool for nonlinear hybrid models. \nThe core algorithm of C2E2 relies on computing reachset over-approximations from validated numerical simulations and what are called {\\em discrepancy functions.\\\/} A discrepancy function for a model bounds the sensitivity of the trajectories of the hybrid system to changes in initial states and inputs. Candidate discrepancy functions can be obtained using a global Lipschitz or using a matrix norm for linear systems.\nHowever, typically these approaches give discrepancy functions that blow-up exponentially with time, and therefore, are not useful for verifying problems with long time horizons.\nThe automatic on-the-fly approach implemented in~\\cite{FanQM0D16:CAV} uses bounds on the Jaobian matrix of the system to get tighter local discrepancy functions and it has been used to verify several benchmark problems. Recently the tool has been extended to handle nonlinear models with dynamics with exponential and trigonometric functions. \n\nFor a (possibly nonlinear) mode with $\\dot x = f(x(t))$, the discrepancy computed by the algorithm of~\\cite{FanQM0D16:CAV} uses the Jacobian matrix $J(x)$ of $f(x)$ and the condition number of $J(x_0)$ evaluated at certain points $x_0$ in the state space. For ill-conditioned matrices, such as what we have in the passive mode, (the $A$-matrix representation of \\eqref{eq:lin1}), the over-approximation error may still blow-up.\nIll-conditioned systems may not only arise from passive dynamics but also from extremely large and small coefficients appearing together in $J(x_0)$.\n\nIn order to address this problem, we have created a MATLAB implementation of C2E2's verification algorithm ({\\bf{\\sf SDVTool\\\/}{}}) for linear models.\nUnlike C2E2, {\\sf SDVTool\\\/}{} \\ does not rely on discrepancy, but instead computes the reachable states under a given linear mode directly.\nThe particular algorithm implemented is the one presented in~\\cite{Duggirala2016}:\nFor an $n$-dimensional system, $n+1$ simulations are performed. From these simulations, special sets called \\emph{generalized star sets}, are generated to represent the exact reachsets. \nFor our purposes, a generalized star set is represented by a pair $\\langle x_0,V \\rangle$, where \n\t$x_0 \\in \\mathbb{R}^n$ is the center state and \n\t$V=\\{v_1,...,v_n \\} \\subseteq \\mathbb{R}^n$ is a standard basis (not necessarily unit vectors), and \n\nthe set defined by $\\langle x_0,V \\rangle$ is \n $$\\{x \\in \\mathbb{R}^n \\ | \\ \\exists \\alpha_1, \\ldots, \\alpha_n \\in [-1,1], x = x_0 + \\sum_{i=1}^n \\alpha_i v_i \\}.$$\nAs reachsets are calculated for time steps, $x_0$ and $V$ are transformed.\nWhen the reachtube from a given mode intersects the guards for a transition, the star sets are aggregated and over-approximated with hyperrectangles. \n If $R_i ^*= $ is the star set reachset obtained at time $t_i$, then the hyperrectangular reachset is: \n\\begin{equation*}\n\\begin{aligned}\nR_i = \\{x \\ | \\ x\\leq x(t_i)+\\sum_{j=1}^n max(-v_j,v_j) \\text{ }\\\\ \\text{and } x\\geq x(t_i)+\\sum_{j=1}^n min(-v_j,v_j) \\}.\n\\end{aligned}\n\\end{equation*}\n\nC2E2 and {\\sf SDVTool\\\/}{} currently accumulates all the reachable sets in ProxA and ProxB that \\emph{may} transition to Passive, and uses their convex hull to begin reachset computations under the Passive mode. It follows that if the time interval during which a transition may occur is large, then the initial set of states under the Passive mode is large, making it very difficult to prove safety. One solution is to allow partitioning and refinement of the initial passive mode set. Since this is not currently implemented in C2E2 or {\\sf SDVTool\\\/}{}, we restrict our experiments to transition interval lengths of 5 minutes or less. \nFor example, checking if the system is safe for a transition $clock\\in [50, 200\\text{ min}]$ could be achieved by running several experiments with small subintervals that cover the original interval.\n\n\n\n\n\n\n\\section*{Acknowledgments}\nWe are grateful for the support of Richard S. Erwin in navigating and modeling the problem presented in this paper, and for Yu Meng's support with the C2E2 experiments. We acknowledge the support of the Air Force Research Laboratory through the Space Scholars Program. \n\n\\bibliographystyle{abbrv}\n\n\\section{Conclusions and future directions}\n\\label{sec:conclusion}\n\n\nIn this case study paper, we present a sequence of linear and nonlinear, nondeterministic benchmark models of autonomous rendezvous between spacecraft with several physical and geometric safety requirements. We designed an LQR controller and verified its safety across the different models, a variety of initial conditions, parameter ranges, and using three different hybrid system verification approaches. The models and requirements are made available online.\n\nThis case study, and in particular the requirement for passive safety, has shed light on the weakness of simulation-driven verification in handling ill-conditioned models.\n\nThe results provide a foundation for verifying more sophisticated maneuvers in future autonomous space operations.\nFor example, we proposed a continuous full-state feedback controller, but it is also possible to consider a situation where full state measurement is not possible and a simple bang-bang controller is required. Control theory tells us that this system maintains marginal stability which implies that errors will never recede, so for reasonably-sized initial sets, the reachable sets may not satisfy tight constraints such as LOS. \n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nA new age of deep space exploration is underway with several ongoing public-private partnerships. A groundwork for a possible mission to Mars in the 2030s is also underway~\\cite{obama_mars}.\nAutonomous operations where a spacecraft can operate independent of human control in a wide variety of conditions are essential for deployment, construction, and maintenance missions in space. \nDespite many spectacular successes like the Mars landing of the Curiosity rover, ensuring safety of autonomous spacecraft operations remains a daunting challenge. The cost of failures can be extreme. \nFor example, NASA's DART spacecraft was designed to rendezvous with the MUBLCOM satellite. In 2005, approximately 11 hours into a 24-hour mission, DART's propellant supply depleted due to excessive use of thrusters, and it began maneuvers for departure. In the process it collided with MUBLCOM; it met only 11 of 27 mission objectives, and the failure resulted in a loss exceeding \\$1 million. \nIn another incident, a navigation error caused the Mars Climate Orbiter to reach as low as 57 kilometers, where it was intended to enter an orbit at an altitude of 140-150 kilometers. The spacecraft was destroyed by the resulting atmospheric stresses and friction and the cost incurred was \\$85 million. \nThese incidents, and several others in~\\cite{wong:sw} highlight the consequences of failures in space applications and demonstrate the need for more rigorous testing before deployment. \n\n\nAlthough formal verification has played an important role in design and safety analysis of spacecraft hardware and software (see, for example~\\cite{Holzmann:2014:MC} and the references therein), they have not been used for model-based design and system-level verification and validation.\nIn this paper, we present and verify a realistic and challenging spacecraft maneuver problem called autonomous rendezvous, proximity operations, and docking (ARPOD). \nThe original hybrid control design problem and its variants are introduced by Jewison and Erwin in~\\cite{erwin-cdc16}. \nARPOD is a fundamental set of operations for a variety of space missions such as on-orbit transfer of personnel~\\cite{Woffinden:rendezvous}, resupply for on-orbit personnel~\\cite{Pinard2007}, assembly~\\cite{Zimpfer:rendezvous}, \nservicing, repair, and refueling~\\cite{Galabova2003}.\n\nThe basic setup for ARPOD consists of a passive module or a {\\em target\\\/} (launched separately into orbit) and a {\\em chaser\\\/} spacecraft that must transport the passive module to an on-orbit assembly location. The chaser maintains a relative bearing measurement to the target, but initially it may be too far away from the target to use its range sensors. Range measurements become available within a given range, giving the chaser accurate relative positioning data so that it can stage itself to dock the target. The target must be docked with a specific angle of approach and closing velocity, so as to avoid collision and ensure that the docking mechanisms on each body will mate. Furthermore, the docking procedure must be completed before the chaser goes into eclipse and loses vision-based sensor data. \nFinally, it is necessary for the system to ensure {\\em passive safety\\\/}. That is, the chaser spacecraft should maintain safe separation from the target even if it loses power and communication during its mission. \n\nIn this paper, we present a suite of hybrid models for the rendezvous portion of the ARPOD mission that can serve as benchmarks for verification tools and serve as building-blocks for more complex operations. \nWe present nonlinear models that consist of nonlinear orbital dynamics ({\\sf NLinProx\\\/}{} and {\\sf NLinProxTh\\\/}{}) and linearized models ({\\sf LinProx\\\/}{} and {\\sf LinProxTh\\\/}{}) using the Clohessy-Wiltshire-Hill (CWH) equations~\\cite{1960}.\nThe rendezvous operation is further subdivided into two phases: Proximity Operations A and B, such that phase A captures an interval of larger ranges than the interval of ranges in phase B. In other words, the chaser enters phase A first and as it moves closer to the target, it enters phase B. After this two-phase rendezvous, the chaser enters a docking phase\/maneuver (i.e. when the chaser is less than 10 meters from its target). We disregard the requirements for this phase for this paper. We develop a switched state-feedback controller using Linear Quadratic Regulation (LQR) for the rendezvous phases. \nThe position-triggered transitions brought about by the switching controller are urgent, resulting in a deterministic hybrid automaton. However, we extend the ARPOD problem in~\\cite{erwin-cdc16} to include the passive safety requirement by introducing a nondeterministic time-triggered transition to the passive mode.\n\nWe have successfully verified the requirements for most of the models using existing hybrid verification tools SpaceEx~\\cite{DBLP:conf\/cav\/FrehseGDCRLRGDM11} and C2E2~\\cite{DuggiralaMVP15,FanQM0D16:CAV},\nand also our new MatLab implementation of a simulation-driven verification algorithm ({\\sf SDVTool\\\/}{}) for linear hybrid models. {\\sf SDVTool\\\/}{} improves C2E2's reachability algorithm, with a new technique~\\cite{Duggirala2016} for obtaining reachsets for linear systems.\nWe obtain verification results for an array of varied initial state configurations and passive transitions times to show the robustness and limits of the switched LQR controller. \nThe experiments, in particular on the passive safety requirement, have demonstrated a weakness in simulation-driven verification approaches in handling ill-conditioned models which suggest a need for further research.\nOverall, we believe that our results and approaches establish feasibility of system-level verification of autonomous space operations, and they provide a foundation for the analysis of more sophisticated maneuvers in the future.\n\n\n\\section{Related work}\n\\label{sec:related}\n\nThere are few academic works on system-level verification of autonomous spacecraft. \nA survey of general verification approaches and how they may apply to small satellite systems is presented in~\\cite{nasa_survey}.\n Architecture and Analysis Design Language (AADL) and verification and validation (V\\&V) over AADL models for satellite systems have been reported in~\\cite{bozzano2010formal}\n\nAn feasibility study for applying formal verification of \nautonomous satellite maneuvers is presented in~\\cite{JGMDE:satellite2012}.\nThat approach relied on creating rectangular abstractions (dynamics of the form $\\dot x \\in [a,b]$) of the satellites dynamics through hybridization and verification using PHAVer~\\cite{Frehse:hscc05} and SpaceEx~\\cite{DBLP:conf\/cav\/FrehseGDCRLRGDM11}. \nThe generated abstract models have simple dynamics but hundreds of locations, and also, the analysis is necessarily conservative. In contrast, the approaches presented in this paper work directly with the linear (nonlinear) hybrid dynamics.\n\nThe ARPOD challenge~\\cite{erwin-cdc16} has been taken up by several researchers in proposing a variety of control strategies. \nA two-stage optimal control strategy is developed in~\\cite{Farahani-cdc16}, where the first part involves trajectory planning under a differentially-flat system and the second part implements Model Predictive Control on a linearized model. \nA supervisor is introduced to robustly coordinate a family of hybrid controllers in~\\cite{sanfelice2016cdc}. \nSafe reachsets (i.e. ReachAvoid sets) are computed for the ARPOD mission in~\\cite{oishi2016cdc} and used to solve for minimum fuel and minimum time trajectories. \n\n\\section{Spacecraft Rendezvous Model}\n\\label{sec:model}\n\nIn this section, we present the detailed development of the hybrid models. First we present the orbital dynamics of the spacecraft in Sections~\\ref{ssec:dynamics}-\\ref{ssec:lindynamics}. Then in Sections~\\ref{sec:hybrid-control}-\\ref{sec:lqr} we present a hybrid controller. Finally, we state the various mission constraints in Section~\\ref{sec:safeSets}. \n\n\\subsection{Nonlinear relative motion dynamics}\n\\label{ssec:dynamics}\n\nThe dynamics of the two spacecraft in orbit---the {\\em target\\\/} and the {\\em chaser\\\/}---are derived from Kepler's laws. We use the simplest case for relative motion in space, where the two spacecraft are restricted to the same orbital plane, resulting in two-dimensional, planar motion. The so called Hill's relative coordinate frame is used. As shown in Figure~\\ref{Hill}, Hill's frame is centered on the target spacecraft, with $+\\hat{\\*i}$-direction pointing radially outward from the Earth, $+\\hat{\\*k}$-direction normal from the orbital plane, and $+\\hat{\\*j}$-direction completing a right-handed system. We further assume that the target moves on a circular orbit, and thus, the $\\hat{\\*j}$-direction aligns with the tangential velocity of the target. \n\nThe restriction on the target's orbit implies that the target-centered frame rotates with constant angular velocity. We will assume the target is in geostationary equatorial orbit (GEO), so its angular velocity is $n = \\sqrt{\\frac{\\mu}{r^3}}$, where $\\mu=3.698\\times 10^{14} m^3\/s^2$ and $r=42164 km$. The chaser's position is represented by the vector $x \\*i + y\\*j $, and the chaser's thrusters provide acceleration in the form of $F_x\\*i + F_y\\*j$. The following equations are derived using Kepler's laws and constitute the nonlinear model of the spacecraft dynamics.\n\\begin{align}\n\\begin{split}\n\\ddot{x} &= n^2 x + 2n\\dot{y} + \\frac{\\mu}{r^2} - \\frac{\\mu}{r_c^3}(r+x) + \\frac{F_x}{m_c}, \\\\ \n\\ddot{y} &= n^2 y - 2n\\dot{x} - \\frac{\\mu}{r_c^3}y + \\frac{F_y}{m_c}, \n\\end{split}\n\\label{eq:nonlin1}\n\\end{align}\nwhere $r_c = \\sqrt{(r+x)^2 + y^2}$ is the distance between the chaser and Earth and $m_c = 500$kg is the mass of the chaser.\n\n\\subsection{Linear dynamics}\n\\label{ssec:lindynamics}\nLinearization of these equations about the system's equilibrium\n point results in the Clohessy-Wiltshire-Hill (CWH) equations \\cite{1960}, which are commonly used to capture the relative motion dynamics of two satellites within a reasonably close range. These equations are:\n\\begin{align}\n\\begin{split}\n\\ddot{x} &= 3n^2 x + 2n\\dot{y} + \\frac{F_x}{m_c}, \\\\\n\\ddot{y} &= - 2n\\dot{x} + \\frac{F_y}{m_c}. \n\\end{split}\n\\label{eq:lin1}\n\\end{align}\n\nLet the state vector be denoted by $\\vec{x} = [x, y, \\dot{x}, \\dot{y}]^{T}$. The state-space form of these linear time-invariant (LTI) equations is:\n$$\\dot{\\vec{x}} = A\\vec{x} + B\\vec{u}, \\ \\text{where},$$\n$$A = \\begin{bmatrix} 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 3n^2 & 0 & 0 & 2n \\\\ 0 & 0 & -2n & 0 \\end{bmatrix}, \nB = \\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\\\ \\frac{1}{m_c} & 0 \\\\ 0 & \\frac{1}{m_c} \\end{bmatrix},\n\\vec{u} = \\begin{bmatrix} F_x \\\\ F_y \\end{bmatrix}.$$\n\n\n\n\\subsection{Hybrid controller model}\n\\label{sec:hybrid-control}\n\nComplete hybrid automaton models for the system with additional documentation are available from~\\href{https:\/\/wiki.illinois.edu\/wiki\/display\/MitraResearch\/Autonomous+Satellite+System+Verification}{from this link}\\footnote{https:\/\/tinyurl.com\/verifysat}. \nVarying ranges of the relative distance between the spacecraft give rise to different constraints and requirements, and therefore, require separate controllers. We present a two-stage hybrid controller for achieving the rendezvous maneuver\\footnote{The rendezvous mission presented in this paper is a subset of the four-stage problem presented in~\\cite{erwin-cdc16}. Our two stages of rendezvous are almost identical to ``Phase 2'' and ``Phase 3'' in~\\cite{erwin-cdc16}.}. We refer to these discrete stages as \\emph{modes}. \nEach discrete mode has an \\emph{invariant} which specifies the conditions under which the system may operate in that mode, which we will first describe in words.\n \n \n\\begin{description}\n\n\\item{{\\em Mode 1\\\/}} or Proximity Operations A (ProxA): the chaser is attempting to rendezvous and its separation distance ($\\rho=\\sqrt{x^2+y^2}$) from the target is in the range 100-1000m.\n\n\\item{{\\em Mode 2}} or Proximity Operations B (ProxB): the chaser is attempting to rendezvous and its separation distance is less than 100m.\n\n\\item{{\\em Mode 3}} or Passive mode: the chaser is no longer attempting to rendezvous and is not using its thrusters, regardless of its separation distance. The system may transition to the Passive mode as a result of a failure or loss of power.\n\\end{description}\n\nThe state of the overall hybrid system is defined by the mode and the valuations of a set of continuous variables: relative position $x$, $y$,\n\tthrusts $F_x$, $F_y$, and a global timer $\\mathit{clock}$.\t\nThere are two timing parameters of the model $t_1$ and $t_2$ that specify the time interval over which the chaser spacecraft may enter the Passive mode.\nWhen the system is in a particular mode, the continuous variables $(x,y)$ evolve according to the (linear or nonlinear) differential equations of the previous section. The thrust inputs $F_x$ and $F_y$ are computed according the full-state feedback controller designed in Section~\\ref{sec:lqr}. \n\n\\begin{figure*}[t!]\n\\centering\n\\subfloat[]{\n\t\\label{hyMod}\n\t\\begin{tikzpicture}[scale=0.45]\n\t\\tikzstyle{every node}=[font=\\scriptsize, circle, font=\\scriptsize, minimum size = 1.6cm,text width=2cm]\n\t\\draw (-7,0) node[fill=blue!20] (p1) {\\center{\\vspace{-0.8cm}ProxA: $$\\dot{\\bar{x}} = (A-BK_1)\\bar{x}$$}};\n\t\\draw (0,0) node[fill=green!20] (p2) {\\center{\\vspace{-0.8cm}ProxB: $$\\dot{\\bar{x}} = (A-BK_2)\\bar{x}$$}};\n\t\\draw (7,0) node[fill=gray!20] (p3) {\\center{\\vspace{-0.8cm}Passive:$$\\dot{\\bar{x}} = A\\bar{x}$$}};\n\t\\tikzstyle{every node}=[font=\\scriptsize,fill=none]\n\t\\draw [ shorten >=1pt,->] (-7,4) to node[left] {$\\bar{x}_0$} (p1);\n\t\\draw [ shorten >=1pt,->] (p1) to [out=45,in=135] node[above] {$\\rho\\leq \\rho_t$} (p2);\n\t\\draw [ shorten >=1pt,->] (p2) to [out=-135,in=-45] node[above=4pt] {$\\rho \\geq \\rho_t$} (p1);\n\t\\draw [ shorten >=1pt,->] (p1) to [out=-50,in=-140] node[above=1pt] {$\\mathit{clock} \\in [t_1,t_2]$} (p3);\n\t\\draw [ shorten >=1pt,->] (p2) to [out=45,in=135] node[above] {$\\mathit{clock} \\in [t_1,t_2]$} (p3);\n\t\\end{tikzpicture}}\n\\subfloat[]{\n\t\\label{rhofig}\n\t\\hspace{5mm}\n\t\\begin{tikzpicture}[scale=0.7]\n\n\t\\fill [blue!20] (-2.4,-2.4) rectangle (2.4,2.4);\n\n\t\\fill [green!20] (0,0) circle (2cm);\n\n\t\\draw [ thick,->] (-2.5,0) to node[right=2cm] {$\\*i$} (2.5,0);\n\t\\draw [ thick,->] (0,-2.5) to node[above=2cm,right] {$\\*j$} (0,2.5);\n\n\t\\draw [shorten >=1pt,->] (0,0) to node[above,right]{$\\rho_t = 100$m} (1.41,1.41);\n\n\t\\draw[thick] (0.83,2) -- (-0.83,2);\n\t\\draw[thick] (0.83,2) -- (2,0.83);\n\t\\draw[thick] (2,0.83) -- (2,-0.83);\n\t\\draw[thick] (2,-0.83) -- (0.83,-2);\n\t\\draw[thick] (-0.83,-2) -- (0.83,-2);\n\t\\draw[thick] (-0.83,-2) -- (-2,-0.83);\n\t\\draw[thick] (-2,-0.83) -- (-2,0.83);\n\t\\draw[thick] (-2,0.83) -- (-0.83,2);\n\t\\end{tikzpicture}}\n\\caption{(a) Hybrid model for spacecraft rendezvous, with linear flow equations shown. The invariants in ProxA and ProxB are defined exclusively by the chaser's position, as shown by corresponding colors in the plane of motion in (b). The transition guards between ProxA and ProxB align exactly with their invariant sets, resulting in urgent transitions. The invariant for Passive mode is $clock > t_1$, irrespective of position. A transition to Passive occurs sometime within an interval of time, and hence is nondeterministic. In (b), the octagon represents how the invariants\/guards are approximated and modeled in the verification tools.}\n\\end{figure*}\n\nWe refer to the time elapsed in the mission with the variable $clock$ but do not consider it an explicit state variable. The invariants in each mode can be more precisely described as $\\rho \\geq 100$ and $\\mathit{clock} \\leq t_2$ for mode 1, $\\rho \\leq 100$ and $\\mathit{clock} \\leq t_2$ for mode 2, and $\\mathit{clock} \\geq t_1$ for mode 3. \nA transition from one mode to another is described by a \\emph{guard}. When the state satisfies the guard condition, the system \\emph{may} take the transition. \nIf a transition is required to occur as soon as possible, this is a called an \\emph{urgent transition}. In this system, the distance-based transitions between modes 1 and 2 are urgent. However, the transitions to mode 3 (Passive mode) are not urgent. There is an interval of time, $\\mathit{clock} \\in[t_1,t_2]$, within which the chaser could nondeterministically transition to the Passive mode. Roughly, a larger $[t_1, t_2]$ interval implies a bigger passive-safety envelope for the mission. These transitions to the Passive mode make the system nondeterministic. Indeed, for some choices of this interval, it is possible for the hybrid system to occupy any one of the three modes at a given time.\n\n\n\\begin{figure*}\n\\centering\n\\begin{minipage}[t]{.47\\textwidth}\n\t\\centering\n\t\\includegraphics[width=.65\\textwidth]{Hillframe.png}\n\t\\caption{Hill's relative coordinate frame. The chaser's relative position vector is $x\\hat{\\*i}+y\\hat{\\*j}$.}\n\t\\label{Hill}\n\\end{minipage}\\qquad\n\\begin{minipage}[t]{.47\\textwidth}\n\t\\centering\n\t\\begin{tikzpicture}[scale=0.8]\n\n\t\\fill [blue!20] (-2.5,-2.5) rectangle (2.5,2.5);\n\n\t\\fill [green!20] (0,0) circle (1.5cm);\n\n\t\\fill [red!60] (0,0) -- (-1.3,0.75) arc (150:210:1.5) -- cycle;\n\n\t\\draw [ thick,->] (-3,0) to node[right=2.5cm] {$\\*i$} (3,0);\n\t\\draw [ thick,->] (0,-3) to node[above=2.5cm,right] {$\\*j$} (0,3);\n\t\n\t\\draw [shorten >=1pt,->] (0,0) to node[above]{$\\rho$} (1,1.33);\n\t\\draw (0,0) circle (1.67cm);\n\t\\draw (-0.87,0.5) arc (150:210:1) node[right] {$60^{\\circ}$};\n\t\\end{tikzpicture}\n\t\\caption{The hybrid model (see Figure~\\ref{hyMod}) captures the chaser's motion in ProxA (blue) and ProxB (green), and we use verification tools to show that whenever $\\rho \\in$ ProxB, the chaser does not leave the LOS region (red).}\n\t\\label{LOSfig}\n\\end{minipage}\n\\end{figure*}\n\n\n\\subsection{Linear Quadratic Control}\n\\label{sec:lqr}\nWe have developed a full-state feedback controller, namely, a Linear Quadratic Regulator (LQR), to drive the chaser towards the target's position. \nClosed-loop feedback is desirable because the system can measure and adjust for errors, and ultimately guarantee liveness (i.e. eventually the target will be reached). LQR is specifically chosen because it is constructed by minimizing a quadratic cost function, which we can choose so as to roughly satisfy our safety constraints. \nLQR is only applicable to linear systems, so we design the control for the linearized model in~\\eqref{eq:lin1}, but we will use the same control with nonlinear dynamics~\\eqref{eq:nonlin1} when applying verification tools.\n\nThe form of an LQR solution is: $\\vec{u} = -K\\vec{x}$, where $K \\in \\mathbb{R}^{4\\times2}$ is a constant matrix and $\\vec{u} = [\\frac{F_x}{m_c}, \\frac{F_y}{m_c}]^T$. The $K$ matrix is found by minimizing the following cost function with respect to $\\vec{u}$: $\\int_{0}^{\\infty} (\\vec{x}^{T}Q\\vec{x}+\\vec{u}^{T}R\\vec{u}) dt $, where $Q$ and $R$ are positive definite matrices.\n\nGiven the form of the control law, we update the definition of the\nmodel given in~\\eqref{eq:lin1} to the following:\n\n\\begin{align}\n\\begin{split}\n\\ddot{x} &= \\left(3n^2-\\frac{k_{11}}{m_c}\\right) x - \\frac{k_{12}}{m_c}y - \\frac{k_{13}}{m_c}\\dot{x} + \\left(2n-\\frac{k_{14}}{m_c}\\right)\\dot{y},\\\\\n\\ddot{y} &= - \\frac{k_{21}}{m_c}x - \\frac{k_{22}}{m_c}y - \\left(2n+ \\frac{k_{23}}{m_c}\\right)\\dot{x} - \\frac{k_{24}}{m_c}\\dot{y}.\n\\label{eq:cl}\n\\end{split}\n\\end{align}\n\nThese equations are the expanded version of the closed-loop form shown in Figure~\\ref{hyMod}. \nLater, we will discuss why we distinguish between the dependence of~\\eqref{eq:lin1} on $\\vec{x}$ and $\\vec{u}$ and~\\eqref{eq:cl} on only $\\vec{x}$.\n\nBryson's method \\cite{brysonrule} is used to help determine an appropriate cost function. Begin with $Q$ and $R$ as diagonal matrices, and choose their values so as to normalize each of the state and input variables. In other words, choose the diagonal elements so that $Q_{ii} = \\frac{1}{max(x_i^2)}$ and $R_{ii} = \\frac{1}{max(u_i^2)}$. Here, the denominators refer to the largest \\emph{desired} value of each variable, which will be determined by the safety constraints and mode invariants. While the LQR gains are obtained with our constraints in mind, the resulting controller does not guarantee these constraints are never violated. This is why further verification is still required.\nThis design process is repeated for modes 1 and 2, and the result is two distinct LQR controllers for each of these modes in our hybrid system. \n\n\n\\subsection{Constraints and safety requirements}\n\\label{sec:safeSets}\nIn this section, we enumerate the properties that define a safe and successful mission, and how they are modeled for verification tools. \n\n\t\\begin{description}\n\t\t\\item[ Thrust constraints]\nDuring the rendezvous stages (ProxA and ProxB), the thrusters cannot provide more than $10 N$ of force in any single direction, therefore, we have the constraints:\n \\[\n |F_x|, |F_y| \\leq 10.\n \\] \n\\item [LOS cone and proximity]\nDuring close-range rendezvous ProxB, the chaser must remain within a line-of-sight (LOS) cone (see Figure \\ref{LOSfig}), and its total velocity must remain under $5$cm\/s, so $\\sqrt{\\dot{x}^2+\\dot{y}^2} \\leq 5$cm\/s. The total velocity constraints cannot be exactly modeled using linear constraints, and a polytopic approximation over $\\dot{x},\\dot{y}$ is used. This is done in the same way as $\\sqrt{x^2+y^2}\\leq \\rho_t$ is approximated (see Figure~\\ref{rhofig}).\n\\item[Separation]\nDuring the Passive mode, the chaser must avoid collision with the target, which is theoretically a point mass at the origin. Even in a theoretical model, a small ball or box should be used to bound this point to account for limitations in numerical precision. In reality, the target satellite's dimensions may range from the order of $1$m\nto $100$m,\nso the size of this bounding box will vary depending on the situation. We use a box with a $0.1$m circumradius. \n\n\\end{description}\n\n\n\\section{Verification results}\n\\label{sec:results}\n\n\nIn this section, we discuss and compare verification results from SpaceEx, C2E2, and our implementation of {\\sf SDVTool\\\/}{}. Based on these results, we reach the broad conclusion that with some manual tweaks, the current hybrid system verification tools are indeed capable of analyzing realistic system-level properties of autonomous spacecraft maneuvers. \n\nIn the following presentation, we pick arbitrary model parameters, but to a large extent our results are robust with respect to parameter variations.\nThat is, the parameters can be tuned to the specific requirements of a real mission. \n\nFor subsequent discussion, we label our models as follows: {\\sf LinProx\\\/}{} denotes the equations in \\eqref{eq:cl}, {\\sf NLinProx\\\/}{} denotes the equations in \\eqref{eq:nonlin1} with the same controller as {\\sf LinProx\\\/}{} substituted into $F_x,F_y$, and {\\sf LinProxTh\\\/}{} denotes a model that will soon be introduced to account for explicit thrust values. \n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\subfloat[]{\\label{mat1:a}\\includegraphics[width=.37\\textwidth]{mat1_pos.png}}%\n\t\\subfloat[]{\\label{space1:a}\\includegraphics[width=.31\\textwidth]{space1_pos.png}}%\n\t\\subfloat[]{\\label{c2e2:a}\\includegraphics[width=.36\\textwidth]{c2e2_pos.png}}\\\\\n\t\\subfloat[]{\\label{mat1:b}\\includegraphics[width=.37\\textwidth]{mat1_vel.png}} %\n\t\\subfloat[]{\\label{space1:b}\\includegraphics[width=.31\\textwidth]{space1_vel.png}}%\n\t\\subfloat[]{\\label{mat3}\\includegraphics[width=.36\\textwidth]{mat3_thrust.png}}%\n\t\\caption{Examples of various generated reachsets. Reachable positions using {\\sf LinProx\\\/}{} in {\\sf SDVTool\\\/}{} shown in (a) and SpaceEx in (b). Reachable velocities using {\\sf LinProx\\\/}{} in {\\sf SDVTool\\\/}{} shown in (d) and SpaceEx in (e). Reachable positions of {\\sf NLinProx\\\/}{} without Passive mode in C2E2 shown in (c). Reachable thrusts using {\\sf LinProxTh\\\/}{} in {\\sf SDVTool\\\/}{} is shown in (f).}\n\t\\label{results}\n\\end{figure*}\n\n\\subsection{Hybrid safety proofs}\nFigure~\\ref{results} shows the typical reachset computations obtained from {\\sf SDVTool\\\/}{}, C2E2, and SpaceEx on the {\\sf LinProx\\\/}, {\\sf LinProxTh\\\/}, and {\\sf NLinProx\\\/} \\ models. These computations also establish the safety of the corresponding systems with respect to the requirements in Section~\\ref{sec:safeSets}. \nOverall, the plots show that the reachsets from the different tools are qualitatively similar. \nFrom the more detailed MatLab plots we can check that no part of the reachable sets intersect with unsafe regions. It is clear from the zoomed in portion of Figure~\\ref{mat1:a} that a reasonably larger collision region would violate safety. \n\nIn C2E2 and SpaceEx, each safety property is loaded and checked individually. In C2E2, the running time for a single property for the nonlinear model {\\sf NLinProx\\\/}{} is in the neighborhood of 5-10 minutes; in SpaceEx, the running time for a single property is on the order of a few seconds. {\\sf SDVTool\\\/}{} checks all (12) properties simultaneously and the running time varies from around 30 seconds to 10 minutes. We do not compare absolute running times in further detail in this paper as each of the tools have different semi-automatic workflows and require widely different execution environments.\n\n\n\n\n\\paragraph{Time horizon} \nTiming is obviously critical for space applications to ensure there is sufficient fuel, but with over-approximated reachability, we can only guarantee the mission is completed within a time upper-bound. This upper-bound obtained from reachability analysis may differ significantly from the actual mission time.\n Therefore, we do not impose any strict completion requirements. Instead, we choose a time horizon that is representative of what we might expect in practice, and focus on observing what behaviors are possible within these limits. Typically, Proximity A operations take 1-5 orbits (at under 4 hours an orbit) and proximity B operations take 45-90 minutes~\\cite{wertz:phases}. We choose a sum of approximately 4.5 hours to be our time horizon.\n\n\\paragraph{Initial states} We calculate a set of initial states assuming that the chaser spacecraft is performing the encompassing mission from~\\cite{erwin-cdc16}. \nWe choose an initial set radius of $[25m,25m,0,0]$\naround the point $\\vec{x}_0 = [-900m,-400m,0m\/s,0m\/s]^T$. This can be interpreted as uncertainty in the chaser's initial position, typically due to loss of precision from sensors and computations, or it can be used to explore multiple initial states of interest. We have successfully verified scenarios with uncertainty in the velocity dimensions as well.\n\n\\paragraph{Unsafe sets} For SpaceEx, C2E2, and {\\sf SDVTool\\\/}{}, we model the safety requirements as a collection of linear inequalities. \nThe LOS cone is approximated with a triangle, so we check three properties to prove the system remains within LOS constraints, and so on. \nMax thrust is effectively a one-dimensional constraint, a nonconvex interval, so two properties will capture the unsafe set for each thrust input (one along $x$-direction, one along $y$-direction). But in order to treat it this way, we must introduce extra variables $u_x,u_y$ to explicitly track the thruster values. These extra variables are the difference between {\\sf LinProx\\\/}{} and {\\sf LinProxTh\\\/}{}.\n\n\\paragraph{Passive transition time} The interval of time during which a transition to the passive mode may occur is trivially bounded by the mission time horizon. For this first example, we choose a small interval at $[120,125\n\\text{min}]$. This ensures that the chaser will operate in mode ProxB before transitioning to the passive mode. \n\n\n\n\\subsection{Adding thrust constraints}\n\\label{sec:6dim}\n In Section~\\ref{sec:safeSets}, we described a constraint on thrust that mimics the physical limitations of our spacecraft. We now set up the 6-dimensional model {\\sf LinProxTh\\\/} \\ so that we can verify this additional requirement. We introduce $u_x,u_y$ as explicit state variables, and solve for their differential equations to obtain:\n\\begin{align}\n\\begin{split}\n\\dot{u}_x &= k_{11}\\dot{x} + k_{12}\\dot{y} + k_{13}\\ddot{x} + k_{14}\\ddot{y},\\\\\n\\dot{u}_y &= k_{21}\\dot{x} + k_{22}\\dot{y} + k_{23}\\ddot{x} + k_{24}\\ddot{y}.\n\\end{split}\n\\label{eq:thrustode}\n\\end{align}\nThere are two equivalent numerical models that will produce different over-approximated reachsets. The first model consists of \\eqref{eq:thrustode} and the following:\n\\begin{align}\n\\begin{split}\n\\ddot{x} &= 3n^2 x + 2n\\dot{y} - \\frac{u_x}{m_c},\\\\\n\\ddot{y} &= - 2n\\dot{x} - \\frac{u_y}{m_c}.\n\\end{split}\n\\label{eq:6dim1}\n\\end{align}\nHere $\\ddot{x}$ and $\\ddot{y}$ account for the effects of thrust inputs by explicitly adding $u_x,u_y$. Since each dimension is over-approximated in the reachset computation and $u_x,u_y$ are functions of position and velocity, the computation for subsequent reachable sets of position and velocity have even more uncertainty. \nRoughly, $u_x,u_y$ act as filters for $x,y,\\dot{x},\\dot{y}$, adding distortion and introducing more uncertainty. \nFigure~\\ref{mat2} shows the effects of these compounding errors. The overarching verification algorithm will partition the initial set to reduce errors stemming from the data structure, but it will have to do this numerous times and may time out in practice.\n\nThe second 6-dimensional model (a variant on {\\sf LinProxTh\\\/}) consists of~\\eqref{eq:cl} and \\eqref{eq:thrustode}. In this case, $\\ddot{x}$ and $\\ddot{y}$ implicitly calculate the thrust, and $u_x,u_y$ are independent ``tracking'' variables. The calculations for $\\ddot{x}$ and $\\ddot{y}$ are equivalent to those in \\eqref{eq:6dim1}, but they bypass the ``filter'' when constructing reachsets. The results are identical to those shown in Figures~\\ref{mat1:a}-\\ref{mat1:b}, with the addition of reachable sets of thrusts shown in Figure~\\ref{mat3}.\n\n Once again, our choice of data structure introduces some uncertainty to the explicit representation of the initial set of $u_x,u_y$ values. This is propagated throughout the analysis.\n\n\nWe use the \\eqref{eq:cl}-\\eqref{eq:thrustode} model to obtain safe thrusting results from {\\sf SDVTool\\\/}{}. The fine reachable sets in Figure~\\ref{mat3} show that the LQR controller operates well within the thrust constraints ($|u_x|,|u_y| \\leq 10N$).\n\n\n\\begin{figure*}\n\\centering\n\\begin{minipage}[t]{.47\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{mat2_pos.png}\n\t\\caption{Coarse overapproximation of the reachable positions, when using the 6-dimensional {\\sf LinProxTh\\\/}{} model from Equation~\\ref{eq:6dim1}.}\n\t\\label{mat2}\n\\end{minipage}\\qquad\n\\begin{minipage}[t]{.47\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{fig2.png}\n\t\\caption{Initial positions (with zero initial velocities) of {\\sf LinProx\\\/}{} that have been verified to be safe. They are safe for Passive transition times up to the time shown by the color map.}\n\t\\label{param1}\n\\end{minipage}\n\\end{figure*}\n\n\n\n\\subsection{Robustness of verification}\nTo demonstrate the robustness of the verification approaches (and the designed controller), we performed several experiments varying the initial set and passive transition times with the {\\sf LinProx\\\/} \\ model. \n\nThe scenarios that guarantee a safe mission are summarized in Figure~\\ref{param1}. \nRoughly, choosing an initial position or subset of positions within the shaded region will result in a safe mission for a transition time or interval within $[0,T]$, where $T$ is the time corresponding to the color at that initial position(s). \nFor these experiments, we consider initial separation between the chaser and the target to be near $1000m$, where this LQR control would start being used. We assume the initial chaser velocity to be zero. \nGenerally, we can conclude that, the closer to the $x$-axis the chaser starts, the later the chaser may safely abort to the passive mode. \nOn the other hand, the neighborhood of states along $\\sim 230^{\\circ}$ are not safe for a passive transition at any time. \n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Basic hypergeometric series and equations}\nThe theory of hypergeometric functions and equations dates back at\nleast as far as Gauss. It has long been and is still an integral\npart of the mathematical literature. In particular, the Galois\ntheory of (generalized) hypergeometric equations attracted the\nattention of many authors. For this issue, we refer the reader to\n\\cite{beukersheckman,beukersbrownheck,katzexpsum} and to the\nreferences therein. We also single out the papers\n\\cite{duvalmitschi,mitschihyperconf}, devoted to the calculation\nof some Galois groups by means of a density theorem (Ramis\ntheorem).\n\nIn this paper we focus our attention on the Galois theory of the\nbasic hypergeometric equations, the later being natural\n$q$-analogues of the hypergeometric equations.\n\nThe \\textit{basic hypergeometric series}\n$\\phi(z)=\\qhyper{a}{b}{c}{z}$ with parameters $(a,b,c) \\in\n(\\mathbb C^*) ^3$ defined by :\n\\begin{eqnarray*}\\qhyper{a}{b}{c}{z}&=&\\sum_{n=0}^{+\\infty}\n\\frac{\\pochamer{a,b}{q}{n} }{\\pochamer{c,q}{q}{n} } z^n\\\\\n&=& \\sum_{n=0}^{+\\infty} \\frac{(1-a)(1-aq)\\cdots\n(1-aq^{n-1})(1-b)(1-bq)\\cdots (1-bq^{n-1})}{(1-q)(1-q^2)\\cdots\n(1-q^{n})(1-c)(1-cq)\\cdots (1-cq^{n-1})} z^n\n\\end{eqnarray*}\nwas first introduced by Heine and was later generalized by\nRamanujan. As regards functional equations, the basic\nhypergeometric series provides us with a solution of the following\nsecond order $q$-difference equation, called the \\textit{basic\nhypergeometric equation} with parameters $(a,b,c)$ :\n\\begin{equation}\\label{equa hypergeo} \\phi(q^2z) - \\frac{(a+b)z-(1+c\/q)}{abz-c\/q} \\phi(qz) + \\frac{z-1}{abz-c\/q}\\phi(z)=0.\\end{equation}\nThis functional equation is equivalent to a functional system.\nIndeed, with the notations :\n$$\\lambda(a,b;c;z)= \\frac{(a+b)z-(1+c\/q)}{abz-c\/q}, \\ \\ \\ \\\n\\mu(a,b;c;z)=\\frac{z-1}{abz-c\/q},$$ a function $\\phi$ is solution\nof $(\\ref{equa hypergeo})$ if and only if the vector\n$\\Phi(z)=\\binom{\\phi(z)}{\\phi(qz)}$ satisfies the functional\nsystem : \\begin{equation}\\label{syst hypergeo} \\Phi(qz) =\n\\qhypermatrice{a}{b}{c}{z} \\Phi(z) \\end{equation} with :\n$$\\qhypermatrice{a}{b}{c}{z}=\\left( \\begin{array}{cc} 0&1\\\\ -\\mu(a,b;c;z) & \\lambda(a,b;c;z) \\end{array}\\right).$$\nThe present paper focuses on the calculation of the Galois group\nof the $q$-difference equation (\\ref{equa hypergeo}) or,\nequivalently, that of the $q$-difference system (\\ref{syst\nhypergeo}). A number of authors have developed $q$-difference\nGalois theories over the past years, among whom Franke\n\\cite{frankepvdifference}, Etingof \\cite{etingofgalois}, Van der\nPut and Singer \\cite{psgaloistheory}, Van der Put and Reversat\n\\cite{prev}, Chatzidakis and Hrushovski \\cite{chatzihru}, Sauloy\n\\cite{sauloyqgaloisfuchs}, Andr\\'e \\cite{andrenoncomm}, etc. The\nexact relations between the existing Galois theories for\n$q$-difference equations are partially understood. For this\nquestion, we refer the reader to \\cite{chatziharsing}, and also to\nour Remark section \\ref{constr}.\n\n\nIn this paper we follow the approach of Sauloy (initiated by\nEtingof in the regular case). Our method for computing the Galois\ngroups of the basic hypergeometric equations is based on a\n$q$-analogue of Schlesinger's density theorem stated and\nestablished in \\cite{sauloyqgaloisfuchs}. Note that some of these\ngroups were previously computed by Hendriks in\n\\cite{hendriksqhyper} using a radically different method\n(actually, the author dealt with the Galois groups defined by Van\nder Put and Singer, but these do coincide with those defined by\nSauloy : see our Remark section \\ref{constr}). On a related topic, \nwe also point out the appendix of \\cite{divizio} which contains the \n$q$-analogue of Schwarz's list. \n\nThe paper is organized as follows. In a first part, we give a\nbrief overview of some results from \\cite{sauloyqgaloisfuchs}. In\na second part, we compute the Galois groups of the basic\nhypergeometric equations in all non-resonant (but possibly\nlogarithmic) cases.\n\n\\section{Galois theory for regular singular $q$-difference equations}\n\nUsing analytic tools together with Tannakian duality, Sauloy\ndeveloped in \\cite{sauloyqgaloisfuchs} a Galois theory for regular\nsingular $q$-difference systems. In this section, we shall first\nrecall the principal notions used in \\cite{sauloyqgaloisfuchs},\nmainly the Birkhoff matrix and the twisted Birkhoff matrix. Then\nwe shall explain briefly that this lead to a Galois theory for\nregular singular $q$-difference systems. Last, we shall state a\ndensity theorem for these Galois groups, which will be of main\nimportance in our calculations.\n\n\\subsection{Basic notions}\\label{section the basic objects}\n\nLet us consider $A \\in \\text{Gl}_n(\\mathbb C(\\{z\\}))$. Following Sauloy\nin \\cite{sauloyqgaloisfuchs}, the $q$-difference system :\n\\begin{equation}\\label{general q diff system}Y(qz)=A(z)Y(z)\\end{equation}\nis said to be \\textit{Fuchsian} at $0$ if $A$ is holomorphic at\n$0$ and if $A(0)\\in \\text{Gl}_n(\\mathbb C)$. Such a system is\nnon-resonant at $0$ if, in addition, $Sp(A(0)) \\cap\nq^{\\mathbb Z^*} Sp(A(0))=\\emptyset$. Last we say that the above\n$q$-difference system is \\textit{regular singular} at $0$ if there\nexists $R^{(0)}\\in \\text{Gl}_n(\\mathbb C(\\{z\\}))$ such that the\n$q$-difference system defined by\n$(R^{(0)}(qz))^{-1}A(z)R^{(0)}(z)$ is Fuchsian at $0$. We have\nsimilar notions at $\\infty$ using the change of variable $z\n\\leftarrow 1\/z$.\n\nIn the case of a global system, that is $A\\in \\text{Gl}_n(\\mathbb C(z))$,\nwe will use the following terminology. If $A\\in\n\\text{Gl}_n(\\mathbb C(z))$, then the system (\\ref{general q diff system})\nis called \\textit{Fuchsian} (resp. \\textit{Fuchsian and non-resonnant}, \\textit{regular\nsingular}) if it is Fuchsian (resp. Fuchsian and non-resonnant,\nregular singular) both at $0$ and at $\\infty$.\n\nFor instance, the basic hypergeometric system (\\ref{syst hypergeo}) is Fuchsian.\\\\\n\n\n\\textit{Local fundamental system of solutions at $0$}. Suppose\nthat (\\ref{general q diff system}) is Fuchsian and non-resonant at\n$0$ and consider $J^{(0)}$ a Jordan normal form of $A(0)$.\nAccording to \\cite{sauloyqgaloisfuchs} there exists $F^{(0)} \\in\n\\text{Gl}_n(\\mathbb C\\{z\\})$ such that :\n\\begin{equation} \\label{transfo jauge}\nF^{(0)}(qz)J^{(0)}=A(z)F^{(0)}(z).\n\\end{equation}\nTherefore, if $e^{(0)}_{J^{(0)}}$ denotes a fundamental system of\nsolutions of the $q$-difference system with constant coefficients\n$X(qz)=J^{(0)}X(z)$, the matrix-valued function\n$Y^{(0)}=F^{(0)}e^{(0)}_{J^{(0)}}$ is a fundamental system of\nsolutions of (\\ref{general q diff system}). We are going to\ndescribe a possible choice for $e^{(0)}_{J^{(0)}}$. We denote by\n$\\theta_q$ the Jacobi theta function defined by\n$\\theta_q(z)=\\pochamer{q}{q}{\\infty}\\pochamer{z}{q}{\\infty}\\pochamer{q\/z}{q}{\\infty}$.\nThis is a meromorphic function over $\\mathbb C^*$ whose zeros are\nsimple and located on the discrete logarithmic spiral\n$q^\\mathbb Z$. Moreover, the functional equation\n$\\theta_q(qz)=-z^{-1}\\theta_q(z)$ holds. Now we introduce, for all\n$\\lambda \\in \\mathbb C^*$ such that $|q| \\leq |\\lambda| < 1$, the\n$q$-character\n$e^{(0)}_{\\lambda}=\\frac{\\theta_q}{\\theta_{q,\\lambda}}$ with\n$\\theta_{q,\\lambda}(z)=\\theta_q(\\lambda z)$ and we extend this\ndefinition to an arbitrary non-zero complex number $\\lambda \\in\n\\mathbb C^*$ requiring the equality\n$e^{(0)}_{q\\lambda}=ze^{(0)}_{\\lambda}$. If\n$D=P\\text{diag}(\\lambda_1,...,\\lambda_n)P^{-1}$ is a semisimple\nmatrix then we set\n$e^{(0)}_{D}:=P\\text{diag}(e^{(0)}_{\\lambda_1},...,e^{(0)}_{\\lambda_n})P^{-1}$.\nIt is easily seen that this does not depend on the chosen\ndiagonalization. Furthermore, consider\n$\\ell_q(z)=-z\\frac{\\theta_q'(z)}{\\theta_q(z)}$ and, if $U$ is a\nunipotent matrix, $e^{(0)}_{U}=\\Sigma_{k=0}^n \\ell_q^{(k)}\n(U-I_n)^k$ with $\\ell_q^{(k)}=\\binom{\\ell_q}{k}$. If\n$J^{(0)}=D^{(0)}U^{(0)}$ is the multiplicative Dunford\ndecomposition of $J^{(0)}$, with $D^{(0)}$ semi-simple and\n$U^{(0)}$ unipotent, we set\n$e^{(0)}_{J^{(0)}}=e^{(0)}_{D^{(0)}}e^{(0)}_{U^{(0)}}$.\\\\\n\n\n\n\\textit{Local fundamental system of solutions at $\\infty$}. Using\nthe variable change $z \\leftarrow 1\/z$, we have a similar\nconstruction at $\\infty$. The corresponding fundamental system of\nsolutions is denoted by\n$Y^{(\\infty)}=F^{(\\infty)}e^{(\\infty)}_{J^{(\\infty)}}$.\\\\\n\nThroughout this section we assume that the system (\\ref{general q\ndiff system}) is global and that it is Fuchsian\nand non-resonant.\\\\\n\n\\textit{Birkhoff matrix}. The linear relations between the two\nfundamental systems of solutions introduced above are given by the\nBirkhoff matrix (also called connection matrix)\n$P=(Y^{(\\infty)})^{-1}Y^{(0)}$. Its entries are elliptic functions\n\\textit{i.e.} meromorphic functions over the elliptic curve\n$\\mathbb{E}_q=\\mathbb C^* \/ q^\\mathbb Z$. \\\\\n\n\\textit{Twisted Birkhoff matrix}. In order to describe a\nZariki-dense set of generators of the Galois group associated to\nthe system (\\ref{general q diff system}), we introduce a\n``twisted\" connection matrix. According to\n\\cite{sauloyqgaloisfuchs}, we choose for all $z \\in \\mathbb C^*$ a\ngroup endomorphism $g_z$ of $\\mathbb C^*$ sending $q$ to $z$.\nBefore giving an explicit example, we have to introduce more\nnotations. Let us, for any fixed $\\tau \\in \\mathbb C$ such that\n$q=e^{-2\\pi i \\tau}$, write $q^y=e^{-2\\pi i \\tau y}$ for all $y\n\\in \\mathbb C$. We also define the (non continuous) function\n$\\log_q$ on the whole punctured complex plane $\\mathbb C^*$ by\n$\\log_q(q^y)=y$ if $y\\in \\mathbb C^* \\setminus \\mathbb R^+$ and we\nrequire that its discontinuity is located just before the cut\n(that is $\\mathbb R^+$) when turning counterclockwise around $0$. We\ncan now give an explicit example of endomorphism $g_z$ namely the\nfunction $g_z: \\mathbb C^*=\\mathbb{U} \\times q^\\mathbb R \\rightarrow\n\\mathbb C^*$ sending $uq^\\omega$ to\n$g_z(uq^\\omega)=z^{\\omega}=e^{-2\\pi i \\tau \\log_q(z) \\omega}$ for\n$(u,\\omega)\\in \\mathbb{U} \\times \\mathbb R$, where $\\mathbb{U}\\subset\n\\mathbb C$ is the unit circle.\n\nThen we set, for all $z$ in $\\mathbb C^*$,\n$\\psi_z^{(0)}(\\lambda)=\\frac{\\qcar{\\lambda}(z)}{g_z(\\lambda)}$ and\nwe define $\\psi_z^{(0)}\\left(D^{(0)}\\right)$, the \\textit{twisted\nfactor} at $0$, by\n$\\psi_z^{(0)}\\left(D^{(0)}\\right)=P\\text{diag}(\\psi_z^{(0)}(\\lambda_1),...,\\psi_z^{(0)}(\\lambda_n))P^{-1}$\nwith $D^{(0)}=P\\text{diag}(\\lambda_1,...,\\lambda_n)P^{-1}$. We\nhave a similar construction at $\\infty$ by using the variable\nchange $z \\leftarrow 1\/z$. The corresponding twisting factor is\ndenoted by $\\psi_z^{(\\infty)}(J^{(\\infty)})$.\n\nFinally, the twisted connection matrix $\\breve{P}(z)$ is :\n\\begin{eqnarray*}\n\\breve{P}(z)&=&\\psi_z^{(\\infty)}\\left(D^{(\\infty)}\\right)P(z)\\psi_z^{(0)}\\left(D^{(0)}\\right)^{-1}.\n\\end{eqnarray*}\n\n\n\n\\subsection{Definition of the Galois groups}\\label{constr}\n\nThe definition of the Galois groups of regular singular\n$q$-difference systems given by Sauloy in\n\\cite{sauloyqgaloisfuchs} is somewhat technical and long. Here we\ndo no more than describe the underlying idea.\\\\\n\n\\textit{(Global) Galois group.} Let us denote by $\\mathcal{E}$ the\ncategory of regular singular $q$-difference systems with\ncoefficients in $\\mathbb C(z)$ (so, the base field is\n$\\mathbb C(z)$; the difference field is $(\\mathbb C(z),f(z) \\mapsto\nf(qz))$). This category is naturally equipped with a tensor\nproduct $\\otimes$ such that $(\\mathcal{E},\\otimes)$ satisfies all\nthe axioms defining a Tannakian category over $\\mathbb C$ except\nthe existence of a \\textit{fiber functor} which is not obvious.\nThis problem can be overcome using an analogue of the\nRiemann-Hilbert correspondance.\n\nThe Riemann-Hilbert correspondance for regular singular\n$q$-difference systems entails that $\\mathcal{E}$ is equivalent to\nthe category $\\mathcal{C}$ of connection triples whose objects are\ntriples $(A^{(0)},P,A^{(\\infty)}) \\in \\text{Gl}_n(\\mathbb C) \\times\n\\text{Gl}_n(\\mathcal{M}(\\mathbb{E}_q)) \\times \\text{Gl}_n(\\mathbb C)$ (we refer to\n\\cite{sauloyqgaloisfuchs} for the complete definition of\n$\\mathcal{C}$). Furthermore $\\mathcal{C}$ can be endowed with a\ntensor product $\\und \\otimes$ making the above equivalence of\ncategories compatible with the tensor products. Let us emphasize\nthat $\\und \\otimes$ is not the usual tensor product for matrices.\nIndeed some twisting factors appear because of the bad\nmultiplicative properties of the $q$-characters $e_{q,c}$ : in\ngeneral $e_{q,c}e_{q,d}\\neq e_{q,cd}$.\n\nThe category $\\mathcal{C}$ allows us to define a Galois group :\n$\\mathcal{C}$ is a Tannakian category over $\\mathbb C$. The functor\n$\\omega_0$ from $\\mathcal{C}$ to $Vect_\\mathbb C$ sending an object\n$(A^{(0)},P,A^{(\\infty)})$ to the underlying vector space\n$\\mathbb C^n$ on which $A ^{(0)}$ acts is a fiber functor. Let us\nremark that there is a similar fiber functor $\\omega_\\infty$ at\n$\\infty$. Following the general formalism of the theory of\nTannakian categories (see \\cite{deligne}), the \\textit{absolute\nGalois group} of $\\mathcal{C}$ (or, using the above equivalence of\ncategories, of $\\mathcal E$) is defined as the pro-algebraic group\n$Aut^{\\und \\otimes}(\\omega_0)$ and the \\textit{global Galois group\nof an object $\\chi$ }of $\\mathcal{C}$ (or, using the above\nequivalence of categories, of an object of $\\mathcal{E}$) is the\ncomplex linear algebraic group $Aut^{\\und\n\\otimes}(\\omega_{0|\\langle \\chi \\rangle})$ where $\\langle \\chi\n\\rangle$ denotes the Tannakian subcategory of $\\mathcal{C}$\ngenerated by $\\chi$. For the sake of simplicity, we will often\ncall $Aut^{\\und \\otimes}(\\omega_{0|\\langle \\chi \\rangle})$ the\n\\textit{Galois group} of $\\chi$\n(or, using the above equivalence of categories, of the corresponding object of $\\mathcal{E}$). \\\\\n\n\\textit{Local Galois groups.} Let us point out that notions of\nlocal Galois groups at $0$ and at $\\infty$ are also available\n(here the difference fields are respectively\n$(\\mathbb C(\\{z\\}),f(z) \\mapsto f(qz) )$ and\n$(\\mathbb C(\\{z^{-1}\\}),f(z) \\mapsto f(qz) )$). As expected, they\nare subgroups of the (global) Galois group. Nevertheless, since\nthese groups are of second importance in what follows, we omit the\ndetails and\nwe refer the interesting reader to \\cite{sauloyqgaloisfuchs}.\\\\\n\n\\noindent \\textbf{Remark.} In \\cite{psgaloistheory}, Van der Put\nand Singer showed that the Galois groups defined using a\nPicard-Vessiot theory can be recovered by means of Tannakian\nduality : it is the group of tensor automorphisms of some suitable\ncomplex valued fiber functor over $\\mathcal E$. Since two complex\nvalued fiber functors on a same Tannakian category are necessarily\nisomorphic, we conclude that\nSauloy's and Van der Put and Singer's theories coincide.\\\\\n\n\nIn the rest of this section we exhibit some natural elements of\nthe Galois group of a given Fuchsian $q$-difference system and we\nstate the density theorem of Sauloy.\n\n\n\\subsection{The density theorem}\n\n\nFix a ``base point\" $y_0\\in \\Omega=\\mathbb C^* \\setminus\n\\{\\text{zeros of } \\det(P(z)) \\text{ or poles of } P(z)\\}$ .\nSauloy exhibits in \\cite{sauloyqgaloisfuchs} the following\nelements of the (global) Galois group associated to the\n$q$-difference system (\\ref{general q diff system}) :\n\\begin{itemize}\n\\item[1.a)] $\\gamma_1(D^{(0)})$ and $\\gamma_2(D^{(0)})$ where :\n$$\\gamma_1:\\mathbb C^*=\\mathbb{U} \\times q^\\mathbb R \\rightarrow \\mathbb{U}$$\nis the projection over the first factor and : $$\\gamma_2 :\n\\mathbb C^*=\\mathbb{U} \\times q^\\mathbb R \\rightarrow \\mathbb C^*$$ is defined by\n$\\gamma_2(uq^\\omega)=e^{2\\pi i \\omega}$.\n\\item[1.b)] $U^{(0)}$.\n\\item[2.a)] $\\breve{P}(y_0)^{-1}\\gamma_1(D^{(\\infty)})\\breve{P}(y_0)$ and $\\breve{P}(y_0)^{-1}\\gamma_2(D^{(\\infty)})\\breve{P}(y_0)$.\n\\item[2.b)] $\\breve{P}(y_0)^{-1} U^{(\\infty)} \\breve{P}(y_0)$.\n\\item[3)] $\\breve{P}(y_0)^{-1}\\breve{P}(z)$, $z \\in \\Omega$. \\\\\n\\end{itemize}\n\nThe following result is due to Sauloy \\cite{sauloyqgaloisfuchs}.\n\n\\begin{theo}\\label{density theo}\nThe algebraic group generated by the matrices 1.a. to 3. is the\n(global) Galois group $G$ of the $q$-difference system\n(\\ref{general q diff system}). The algebraic group generated by\nthe matrices 1.a) and 1.b) is the local Galois group at $0$ of the\n$q$-difference system (\\ref{general q diff system}). The algebraic\ngroup generated by the matrices 2.a) and 2.b) is the local Galois\ngroup at $\\infty$ of the $q$-difference system (\\ref{general q\ndiff system}).\n\\end{theo}\n\nThe algebraic group generated by the matrices 3) is called the\n\\textit{connection component} of the Galois group $G$. The\nfollowing result is easy but very useful. Its proof is left to the\nreader.\n\n\\begin{lem} The connection component of the Galois group $G$ of a regular singular $q$-difference system is a subgroup of the\nidentity component $G^I$ of $G$.\n\\end{lem}\n\n\\section{Galois groups of the basic hypergeometric equations : non-resonant and non-logarithmic cases}\\label{section generique}\n\nWe write $a=uq^\\alpha$, $b=vq^\\beta$ and $c=wq^\\gamma$ with\n$u,v,w\\in\\mathbb{U}$ and $\\alpha,\\beta,\\gamma \\in \\mathbb R$ (we\nchoose a logarithm of $q$).\\\\\n\nIn this section we are aiming to compute the Galois group of the\nbasic hypergeometric system (\\ref{syst hypergeo}) under the\nfollowing assumptions :\n$$a\/b \\not \\in q^\\mathbb Z \\text{ and }c \\not \\in q^\\mathbb Z.$$\n\nFirst, we give explicit formulas for the generators of the Galois group of (\\ref{syst hypergeo}) involved in Theorem \\ref{density theo}.\\\\\n\n\\textit{Local fundamental system of solutions at $0$.} We have :\n$$\\qhypermatrice{a}{b}{c}{0}=\\pmatrice{1}{1}{1}{q\/c}\\pmatrice{1}{0}{0}{q\/c} \\pmatrice{1}{1}{1}{q\/c}^{-1}.$$\nHence the system (\\ref{syst hypergeo}) is non-resonant, and\nnon-logarithmic at $0$ since $\\qhypermatrice{a}{b}{c}{0}$ is\nsemi-simple. A fundamental system of solutions at $0$ of\n(\\ref{syst hypergeo}) as described in section \\ref{section the\nbasic objects} is given by\n$\\yz{a}{b}{c}{z}=\\fz{a}{b}{c}{z}e^{(0)}_{\\jz{c}}(z)$ with $\\jz{c}=\n\\text{diag}(1,q\/c)$ and :\n$$\\fz{a}{b}{c}{z}=\\pmatrice{\\qhyper{a}{b}{c}{z}}{\\qhyper{aq\/c}{bq\/c}{q^2\/c}{z}}\n{\\qhyper{a}{b}{c}{qz}}{(q\/c) \\qhyper{aq\/c}{bq\/c}{q^2\/c}{qz}}.$$\\\\\n\n\\textit{Generators of the local Galois group at $0$.} We have two generators :\n$$\\pmatrice{1}{0}{0}{e^{2\\pi i \\gamma}} \\text{ and }\n\\pmatrice{1}{0}{0}{w}.$$\n\n\\textit{Local fundamental system of solutions at $\\infty$.} We\nhave :\n$$\\qhypermatrice{a}{b}{c}{\\infty}=\\pmatrice{1}{1}{1\/a}{1\/b} \\pmatrice{1\/a}{0}{0}{1\/b} \\pmatrice{1}{1}{1\/a}{1\/b}^{-1}.$$\nHence the system (\\ref{syst hypergeo}) is non-resonant and\nnon-logarithmic at $\\infty$ and a fundamental system of solutions\nat $\\infty$ of (\\ref{syst hypergeo}) as described in section\n\\ref{section the basic objects} is given by\n$\\yinf{a}{b}{c}{z}=\\finf{a}{b}{c}{z}e^{(\\infty)}_{\\jinf{a}{b}}(z)$\nwith $\\jinf{a}{b}= \\text{diag}(1\/a,1\/b)$ and :\n$$\\finf{a}{b}{c}{z}=\n\\pmatrice{\\qhyper{a}{aq\/c}{aq\/b}{\\frac{cq}{ab}z^{-1}}}{\\qhyper{b}{bq\/c}{bq\/a}{\\frac{cq}{ab}z^{-1}}}\n{\\frac{1}{a} \\qhyper{a}{aq\/c}{aq\/b}{\\frac{c}{ab}z^{-1}}}{\\frac{1}{b} \\qhyper{b}{bq\/c}{bq\/a}{\\frac{c}{ab}z^{-1}}}.$$\\\\\n\n\\textit{Generators of the local Galois group at $\\infty$.} We have two generators :\n$$\\breve{P}(y_0)^{-1}\\pmatrice{e^{2\\pi i \\alpha}}{0}{0}{e^{2\\pi i \\beta}}\\breve{P}(y_0) \\text{ and } \\breve{P}(y_0)^{-1}\\pmatrice{u}{0}{0}{v}\\breve{P}(y_0).$$\\\\\n\n\\textit{Birkhoff matrix.} The Barnes-Mellin-Watson formula (\\textit{cf.} \\cite{gasperrahman}) entails that :\n$$P(z)= (e^{(\\infty)}_{\\jinf{a}{b}}(z))^{-1}\n\\pmatrice{\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}} \\frac{\\theta_q (a z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}} \\frac{\\theta_q (\\frac{aq}{c} z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}} \\frac{\\theta_q (b z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}} \\frac{\\theta_q (\\frac{bq}{c} z)}{\\theta_q(z)}}\ne^{(0)}_{\\jz{c}}(z).$$\\\\\n\n\\textit{Twisted Birkhoff matrix.} We have :\n$$\\breve{P}(z)= \\pmatrice{(1\/z)^{-\\alpha}}{0}{0}{(1\/z)^{-\\beta}}\n\\pmatrice{\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{aq}{c} z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c} z)}{\\theta_q(z)}}\n\\pmatrice{1}{0}{0}{z^{1-\\gamma}}.$$\n\n$_{}$\\vskip 10 pt\n\nWe need to consider different cases.\n\n$_{}$\\vskip 10 pt\n\n\\condun{1} $\\underline{a,b,c,a\/b,a\/c, b\/c \\not \\in q^\\mathbb Z\n\\text{ and } a\/b \\text{ or } c \\not \\in \\pm q^{\\mathbb Z\/2}}$.\n\n$_{}$\\vskip 5 pt\n\nUnder this assumption the four numbers $\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}$,\n$\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}$,\n$\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}$ and\n$\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}$ are non-zero.\n\n\\begin{prop}\nSuppose that \\condun{1} holds. Then the natural action of $G^I$ on $\\mathbb C^2$ is irreducible.\n\\end{prop}\n\n\\begin{proof}\nSuppose, at the contrary, that the action of $G^I$ is reducible\nand let $L \\subset \\mathbb C^2$ be an invariant line.\n\nRemark that $L$ is distinct from $\\mathbb C \\vect{1}{0}$ and\n$\\mathbb C \\vect{0}{1}$. Indeed, assume at the contrary that\n$L=\\mathbb C \\vect{1}{0}$ (the case $L=\\mathbb C \\vect{0}{1}$ is\nsimilar). The line $L=\\mathbb C \\vect{1}{0}$ being in particular\ninvariant by the connection component, we see that the line\ngenerated by $\\breve{P}(z) \\vect{1}{0}$ does not depend on $z \\in\n\\Omega$. This yields a contradiction because the ratio of the\ncomponents of $\\breve{P}(z) \\vect{1}{0}=\n\\vect{\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}(1\/z)^{-\\beta}}$ depends on $z$\n(remember that $a\/b \\not \\in q^\\mathbb Z$).\n\nOn the other hand, since, for all $n\\in\\mathbb N$, both matrices\n$\\pmatrice{1}{0} {0}{e^{2\\pi i \\gamma n}}$ and\n$\\pmatrice{1}{0}{0}{w^n}$ belong to $G$ and since $G^I$ is a\nnormal subgroup of $G$, both lines $L_n:=\\pmatrice{1}{0}\n{0}{e^{2\\pi i \\gamma n}}L$ and $L'_n:=\\pmatrice{1}{0}{0}{w^n} L$\nare also invariant by $G^I$.\n\n\nNote that because \\condun{1} holds, at least one of the complex\nnumbers $w,e^{2\\pi i \\gamma},u\/v,e^{2\\pi i (\\alpha-\\beta)}$ is\ndistinct from $\\pm 1$.\n\nFirst, suppose that $w \\neq \\pm 1$. We have seen that $L \\neq\n\\mathbb C \\vect{1}{0}, \\mathbb C \\vect{0}{1}$, hence $L_0,L_1,L_2$\nare three distinct lines invariant by the action of $G^I$. This\nimplies that $G^I$ consists of scalar matrices : this is a\ncontradiction (because, for instance, $\\mathbb C \\vect{1}{0}$ is\nnot invariant for the action of $G^I$). Hence, if $w \\neq \\pm 1$\nwe have proved that $G^I$ acts irreducibly.\n\nThe case $e^{2\\pi i \\gamma} \\neq \\pm 1$ is similar.\n\nLast, the proof is analogous if $u\/v \\neq \\pm 1$ or $e^{2\\pi i\n(\\alpha-\\beta)} \\neq \\pm 1$ (we then use the fact that, for all\n$z\\in \\Omega$, $G^I$ is normalized by $\\breve{P}(z)^{-1}\n\\pmatrice{u}{0}{0}{v} \\breve{P}(z)$ and $\\breve{P}(z)^{-1}\n\\pmatrice{e^{2\\pi i \\alpha}}{0}{0}{e^{2\\pi i \\beta}} \\breve{P}(z)$\nand that there exists $z \\in \\Omega$ such that $\\breve{P}(z)L$ is\ndistinct from $\\mathbb C \\vect{1}{0}$ and $\\mathbb C \\vect{0}{1}$).\n\\end{proof}\n\nWe have the following theorem.\n\n\\begin{theo}\\label{theo un}\nSuppose that \\condun{1} holds.\nThen we have the following dichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $abq\/c \\not \\in q^\\mathbb Z$ then $G=\\text{Gl}_2(\\mathbb C)$;\n\\item[$\\bullet$] if $abq\/c \\in q^\\mathbb Z$ then $G=\\overline{\\langle \\text{Sl}_2(\\mathbb C),\\sqrt{w}I,e^{\\pi i \\gamma}I \\rangle}$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nSince $G^I$ acts irreducibly on $\\mathbb C^2$, the general theory\nof algebraic groups entails that $G^I$ is generated by its center\n$Z(G^I)$ together with its derived subgroup $G^{I,der}$ and that\n$Z(G^I)$ acts as scalars. Hence, $G^{I,der} \\subset\n\\text{Sl}_2(\\mathbb C)$ also acts irreducibly on $\\mathbb C^2$. Therefore\n$G^{I,der}=\\text{Sl}_2(\\mathbb C)$ (a connected algebraic group of\ndimension less than or equal to $2$ is solvable hence $\\text{dim}\n(G^{I,der}) =3$ and $G^{I,der}=\\text{Sl}_2(\\mathbb C)$). In order to\ncomplete the proof, it is sufficient to determine $\\text{det}(G)$.\nWe have :\n\\begin{eqnarray*}\n\\text{det}(\\breve{P}(z)\n&=&\\frac{(1\/z)^{-(\\alpha+\\beta)}z^{1-\\gamma}}{\\pochamer{q^2\/c,a\/b,c,b\/a}{q}{\\infty}}\n\\left(\\underbrace{\\theta_q(b)\\theta_q(c\/a) \\frac{\\theta_q (a\nz)}{\\theta_q(z)} \\frac{\\theta_q (\\frac{bq}{c} z)}{\\theta_q(z)}\n-\\theta_q(c\/b)\\theta_q(a)\\frac{\\theta_q (\\frac{aq}{c}\nz)}{\\theta_q(z)} \\frac{\\theta_q (b\nz)}{\\theta_q(z)}}_{\\psi(z)}\\right).\n\\end{eqnarray*}\nA straightforward calculation shows that the function :\n$$\\theta_q(b)\\theta_q(c\/a) \\theta_q (a z) \\theta_q (\\frac{bq}{c}\nz) -\\theta_q(c\/b)\\theta_q(a)\\theta_q (\\frac{aq}{c} z)\\theta_q (b\nz)$$ vanishes for $z \\in q^\\mathbb Z$ and for $z \\in\n\\frac{c}{abq}q^\\mathbb Z$. On the other hand $\\psi$ is a solution\nof the first order $q$-difference equation $y(qz)=\\frac{c}{abq}\ny(z)$. Hence, if we suppose that $abq\/c \\not \\in q^\\mathbb Z$, we\ndeduce that the ratio $\\chi(z)=\\frac{\\psi(z)}{\\frac{\\theta_q\n(\\frac{abq}{c} z)}{\\theta_q(z)}}$ defines an holomorphic elliptic\nfunction over $\\mathbb C^*$. Therefore $\\chi$ is constant and,\nevaluating $\\chi$ at $z=1\/b$,\n we get :\n$$\\chi=-b \\theta_q(a\/b)\n \\theta_q (c).$$\nFinally, we obtain the identity :\n\\begin{eqnarray} \\label{det}\n\\text{det}(\\breve{P}(z))&=&\\frac{1-q\/c}{1\/a-1\/b}(1\/z)^{-(\\alpha+\\beta)}z^{1-\\gamma}\n\\frac{\\theta_q (\\frac{abq}{c} z)}{\\theta_q(z)}.\n\\end{eqnarray}\nBy analytic continuation (with respect to the parameters) we see\nthat this formula also holds if $abq\/c \\in q^\\mathbb Z$.\n\nConsequently, if $abq\/c \\not \\in q^\\mathbb Z$, for any fixed $y_0\n\\in \\Omega$, $\\text{det}(\\breve{P}(y_0)^{-1}\\breve{P}(z))$ is a non\nconstant holomorphic function (with respect to $z$). This implies\nthat $G=G^I=\\text{Gl}_2(\\mathbb C)$. On the other hand, if $abq\/c \\in\nq^\\mathbb Z$, then we have that\n$\\text{det}(\\breve{P}(y_0)^{-1}\\breve{P}(z))=1$, so that the\nconnection component of the Galois group is a subgroup of\n$\\text{Sl}_2(\\mathbb C)$ and the Galois group $G$ is the smallest\nalgebraic group which contains $\\text{Sl}_2(\\mathbb C)$ and $\\{\n\\sqrt{w}I,e^{\\pi i \\gamma}I\\}$.\n\\end{proof}\n\n\nWe are going to study the case $a,b,c,a\/b,a\/c, b\/c \\not \\in q^\\mathbb Z$ and $a\/b,c \\in \\pm q^{\\mathbb Z+1\/2}$ in two steps.\\\\\n\n$_{}$\\vskip 10 pt\n\n\\condun{2} $\\underline{a,b,c,a\/b,a\/c, b\/c \\not \\in q^\\mathbb Z\n\\text{ and } q^\\mathbb Z a \\cup q^\\mathbb Z b \\cup q^\\mathbb Z\naq\/c \\cup q^\\mathbb Z bq\/c=q^\\mathbb Z a \\cup -q^\\mathbb Z a \\cup\nq^{\\mathbb Z+1\/2} a \\cup -q^{\\mathbb Z+1\/2} a}$\n\n$_{}$\\vskip 5 pt\n\n We first establish a preliminary result.\n\n\\begin{lem}\\label{hg}\nSuppose that \\condun{2} holds. Then any functional equation of the\nform $Az^{n\/2}\\theta_q(q^Naz)+Bz^{m\/2}\\theta_q(-q^M\naz)+Cz^{l\/2}\\theta_q(q^Lq^{1\/2}az)+Dz^{k\/2}\n\\theta_q(-q^Kq^{1\/2}az)=0$ with $A,B,C,D \\in \\mathbb C$,\n$n,m,l,k,N,M,L,K\\in \\mathbb Z$ is trivial, that is $A=B=C=D=0$.\n\\end{lem}\n\n\\begin{proof}\nUsing the non-trivial monodromy of $z^{1\/2}$, we reduce the\nproblem to the case of $n,m,l,k$ being odd numbers. In this case,\nusing the functional equation $\\theta_q(qz)=-z^{-1}\\theta_q(z)$,\nwe can assume without loss of generality that $n=l=m=k=0$. The\nexpansion of $\\theta_q$ as an infinite Laurent series\n$\\theta_q(z)=\\sum_{j\\in\\mathbb Z}q^{\\frac{j(j-1)}{2}}(-z)^j$\nensures that, for all $j \\in \\mathbb Z$, the following equality\nholds :\n$$A(q^{N})^j+B(-q^{M})^j+C(q^{L+1\/2})^j+D(-q^{K+1\/2})^j=0.$$ Considering the associated generating series, this implies that :\n$$\\frac{A}{1-q^Nz}+\\frac{B}{1+q^Mz}+\\frac{C}{1-q^{L+1\/2}z}+\\frac{D}{1+q^{K+1\/2}z}=0.$$\nHence, considering the poles of this rational fraction, we obtain\n$A=B=C=D=0$.\n\\end{proof}\n\n\n\n\\begin{prop}\nSuppose that \\condun{2} holds. Then the natural action of $G^I$ on $\\mathbb C^2$ is irreducible.\n\\end{prop}\n\n\\begin{proof}\nSuppose, at the contrary, that the action of $G^I$ is reducible\nand consider an invariant line $L\\subset \\mathbb C^2$. In\nparticular, $L$ is invariant under the action of the connection\ncomponent. Consequently, the line $\\breve{P}(z)L$ does not depend\non $z\\in \\Omega$. This is impossible using Lemma \\ref{hg} (the cases\n$L=\\mathbb C \\vect{1}{0}$ or $\\mathbb C \\vect{0}{1}$ are excluded by\ndirect calculation; for the remaining cases consider the ratio of\nthe coordinates of a generator of $L$ and apply Lemma \\ref{hg}).\nWe get a contraction, hence prove that $G^I$ acts irreducibly.\n\\end{proof}\n\n\\begin{theo}\nIf \\condun{2} holds then we have the following dichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $abq\/c \\not \\in q^\\mathbb Z$ then $G=\\text{Gl}_2(\\mathbb C)$;\n\\item[$\\bullet$] if $abq\/c \\in q^\\mathbb Z$ then $G=\\overline{\\langle \\text{Sl}_2(\\mathbb C),\\sqrt{w}I,e^{\\pi i \\gamma}I \\rangle}$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nThe proof follows the same lines as that of theorem \\ref{theo un}.\n\\end{proof}\n\n\nThe remaining subcases are $b\\in -aq^\\mathbb Z$ and $c\\in\n-q^\\mathbb Z$; $b\\in -aq^{\\mathbb Z+1\/2}$ and $c\\in\n-q^{\\mathbb Z+1\/2}$; $b\\in\naq^{\\mathbb Z+1\/2}$ and $c\\in q^{\\mathbb Z+1\/2}$.\\\\\n\n$_{}$\\vskip 10 pt\n\n\\condun{3} $\\underline{a,b,c,a\/b,a\/c, b\/c \\not \\in q^\\mathbb Z\n\\text{ and } b\\in -aq^\\mathbb Z \\text{ and } c\\in -q^\\mathbb Z}$.\n\n$_{}$\\vskip 5 pt\n\n We use the following notations : $b = -aq^\\delta\n\\text{ and } c = -q^\\gamma$ with\n$\\delta=\\beta-\\alpha,\\gamma\\in\\mathbb Z$.\n\nThe twisted connection matrix takes the following form :\n\\begin{eqnarray*}\n\\breve{P}(z) &=&(1\/z)^{-\\alpha} \\pmatrice{\n\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}}\n {\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n \\frac{\\theta_q (\\frac{aq}{c} z)}{\\theta_q(z)}z^{1-\\gamma}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}z^{\\delta} }\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}z^{1+\\delta-\\gamma}} \\\\\n&=&(1\/z)^{-\\alpha} \\pmatrice{\n\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}} {\n\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n q^{\\frac{\\gamma(1-\\gamma)}{2}}a^{\\gamma-1} \\frac{\\theta_q (-a z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\nq^{\\frac{-\\delta(\\delta-1)}{2}}a^{-\\delta} \\frac{\\theta_q (-a\nz)}{\\theta_q(z)}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\nq^{-\\frac{(\\delta-\\gamma +1)(\\delta -\\gamma)}{2}}(-a)^{\\gamma -\\delta -1}\\frac{\\theta_q (a\nz)}{\\theta_q(z)}}\n\\end{eqnarray*}\n\n\\begin{theo}\nSuppose that \\condun{3} holds. We have $G=R\n\\pmatrice{\\mathbb C^*}{0}{0}{\\mathbb C^*} R^{-1} \\cup\n\\pmatrice{1}{0}{0}{-1} R \\pmatrice{\\mathbb C^*}{0}{0}{\\mathbb C^*}\nR^{-1}$ for some $R\\in \\text{Gl}_2(\\mathbb C)$ of the form\n$\\pmatrice{1}{1}{C}{-C}$, $C\\in\\mathbb C^*$.\n\\end{theo}\n\n\\begin{proof}\nRemark that there exist two nonzero constants $A,B$ such that, for\nall $z\\in \\Omega$ :\n\\begin{eqnarray*}\n\\breve{P}(-1\/a)^{-1}\\breve{P}(z)&=&(-a)^\\alpha \\frac{\\theta_q\n(-1\/a)}{\\theta_q(-1)} (1\/z)^{-\\alpha} \\pmatrice{\\frac{\\theta_q (a\nz)}{\\theta_q(z)}} {A \\frac{\\theta_q (-a z)}{\\theta_q(z)}} {B\n\\frac{\\theta_q (-a z)}{\\theta_q(z)} } {\\frac{\\theta_q (a\nz)}{\\theta_q(z)}}\\\\\n&=&(-a)^\\alpha \\frac{\\theta_q (-1\/a)}{\\theta_q(-1)} (1\/z)^{-\\alpha} R\n\\pmatrice{\\frac{\\theta_q (a z)}{\\theta_q(z)} + \\sqrt{BA}\n\\frac{\\theta_q (-a z)}{\\theta_q(z)}}{0} {0} {\\frac{\\theta_q (a\nz)}{\\theta_q(z)} - \\sqrt{BA} \\frac{\\theta_q (-a z)}{\\theta_q(z)}}\nR^{-1}\n\\end{eqnarray*}\nwith $R=\\pmatrice{1}{1}{\\sqrt{B\/A}}{-\\sqrt{B\/A}}$.\n\nFurthermore, we claim that the functions $X(z):=(1\/z)^{-\\alpha}\n(\\frac{\\theta_q (a z)}{\\theta_q(z)} + \\sqrt{BA} \\frac{\\theta_q (-a\nz)}{\\theta_q(z)})$ and $Y(z):=(1\/z)^{-\\alpha} (\\frac{\\theta_q (a\nz)}{\\theta_q(z)} - \\sqrt{BA} \\frac{\\theta_q (-a z)}{\\theta_q(z)})$\ndo not satisfy any non-trivial relation of the form $X^rY^s=1$\nwith $(r,s)\\in\\mathbb Z^2 \\setminus \\{(0,0\\}$. Indeed, suppose on\nthe contrary that such a relation holds. Then\n$\\frac{((1\/z)^{-\\alpha} (\\theta_q (a z) + \\sqrt{BA} \\theta_q (-a\nz)))^r}{((1\/z)^{-\\alpha} (\\theta_q (a z) - \\sqrt{BA} \\theta_q (-a\nz)))^s}=\\theta_q(z)^{s-r}$. Let us first exclude the case $r\\neq\ns$. If $s>r$ then we conclude that $\\theta_q (a z) + \\sqrt{BA}\n\\theta_q (-a z)$ must vanish on $q^\\mathbb Z$. In particular,\n$\\theta_q (a) + \\sqrt{BA} \\theta_q (-a)=0$ and $\\theta_q (aq) +\n\\sqrt{BA} \\theta_q (-aq) = -(az)^{-1}(\\theta_q (a z) - \\sqrt{BA}\n\\theta_q (-a z))=0$, so $\\theta_q (a)=0$ that is $a\\in\nq^\\mathbb Z$. This yields a contradiction. The case $r>s$ is\nsimilar by symmetry. Hence we have $r=s$ so that\n$\\left(\\frac{\\theta_q (a z) + \\sqrt{BA} \\theta_q (-a z)}{\\theta_q\n(a z) - \\sqrt{BA} \\theta_q (-a z)}\\right)^r=1$. Therefore the\nfunction $\\frac{\\theta_q (a z) + \\sqrt{BA} \\theta_q (-a\nz)}{\\theta_q (a z) - \\sqrt{BA} \\theta_q (-a z)}$ is constant. This\nis clearly impossible and our claim is proved.\n\n\n\n\nThis ensures that the connection component of $G^I$, generated by\nthe matrices $\\breve{P}(-1\/a)^{-1}\\breve{P}(z)$, $z \\in \\Omega$, is\nequal to $R\\pmatrice{\\mathbb C^*}{0}{0}{\\mathbb C^*}R^{-1}$.\nConsequently, $G$ is generated as an algebraic group by\n$R\\pmatrice{\\mathbb C^*}{0}{0}{\\mathbb C^*}R^{-1}$,\n$\\pmatrice{1}{0}{0}{-1}$,\n$\\breve{P}(-1\/a)^{-1}\\pmatrice{u}{0}{0}{-u}\\breve{P}(-1\/a)=\\pmatrice{u}{0}{0}{-u}$\nand $\\breve{P}(-1\/a)^{-1}\\pmatrice{e^{2\\pi i\n\\alpha}}{0}{0}{e^{2\\pi i \\alpha}}\\breve{P}(-1\/a)=\\pmatrice{e^{2\\pi\ni \\alpha}}{0}{0}{e^{2\\pi i \\alpha}}$. The theorem follows.\n\\end{proof}\n\n$\\bullet$ Both cases ($b\\in -aq^{\\mathbb Z+1\/2}$ and $c\\in -q^{\\mathbb Z+1\/2}$) and ($b\\in\naq^{\\mathbb Z+1\/2}$ and $c\\in q^{\\mathbb Z+1\/2}$) are similar.\\\\\n\n$_{}$\\vskip 10 pt\n\n\\condun{4} $\\underline{a \\in q^{\\mathbb N^*}}$.\n\n$_{}$\\vskip 5 pt\n\nIn this case, the twisted connection matrix is lower triangular :\n\n\\begin{eqnarray*}\n\\breve{P}(z) &=&\\pmatrice{\n\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n(-1)^{\\alpha} q^{-\\frac{\\alpha (\\alpha -1)}{2}}} {0}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}(1\/z)^{-\\beta}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}}\n\\end{eqnarray*}\n\n\\begin{theo}\\label{theo theo}\nSuppose that \\condun{4} holds.\nWe have the following trichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $b\/c \\not\\in q^\\mathbb Z$ then\n$G=\\pmatrice{1}{0} {\\mathbb C}{\\mathbb C^*}$;\n\\item[$\\bullet$] if $c\/b \\in q^{\\mathbb N^*}$ then\n$G=\\pmatrice{1}{0}{\\mathbb C}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$;\n\\item[$\\bullet$] if $bq\/c \\in q^{\\mathbb N^*}$ then\n$G=\\pmatrice{1}{0} {0}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nRemark that in each case there exist two constants $A,B$ with $B\n\\neq 0$ such that, for all $z\\in \\Omega$,\n$\\breve{P}(1\/b)^{-1}\\breve{P}(z)= \\pmatrice{1}{0}{A\\frac{\\theta_q\n(b z)}{\\theta_q(z)}(1\/z)^{-\\beta}}{B \\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}}$. Hence, the\nconnection component is a subgroup of\n$\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$.\n\nAssume $b\/c \\not\\in q^\\mathbb Z$. Then $A\\neq 0$ and we claim that\nthe connection component is equal to\n$\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$. Indeed, for all\n$n\\in\\mathbb Z$, the following matrix :\n$$(\\breve{P}(1\/b)^{-1}\\breve{P}(z))^n=\\pmatrice{1}{0}{A\\frac{\\theta_q\n(b z)}{\\theta_q(z)}(1\/z)^{-\\beta}\\frac{1-\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n}{1-B\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}}}{\\left(B\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n}$$ belongs to\nthe connection component. Consider a polynomial in two variables\n$K(X,Y) \\in \\mathbb C[X,Y]$ such that :\n$$K(A\\frac{\\theta_q (b\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}\\frac{1-\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n}{1-B\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}},\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n)=0.$$ If $K$\nwas non zero then we could assume that $K(X,0)\\neq 0$. But, for\nall $z\\in\\Omega$ in a neighborhood of $\\frac{c}{bq}$, we have\n$\\left|B \\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right|<1$, hence\nletting $n$ tend to $+\\infty$, we would get $K(A\\frac{\\theta_q (b\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}\\frac{1}{1-B \\frac{\\theta_q\n(\\frac{bq}{c} z)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}},0)=0$\nwhich would imply $K(X,0)=0$. This proves that $K=0$. In other\nwords the only algebraic subvariety of $\\mathbb C \\times\n\\mathbb C^*$ containing $(A\\frac{\\theta_q (b\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}\\frac{1-\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n}{1-B\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}},\\left(B \\frac{\\theta_q\n(\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}\\right)^n)$ for all\n$n\\in\\mathbb Z$ is $\\mathbb C \\times \\mathbb C^*$ itself. In\nparticular, the algebraic group generated by the matrix\n$(\\breve{P}(1\/b)^{-1}\\breve{P}(z))^n$ for all $n\\in\\mathbb Z$ is\n$\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$, hence the connection\ncomponent is equal to $\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$.\n\nIt is now straightforward that\n$G=\\pmatrice{1}{0}{\\mathbb C}{\\mathbb C^*}$.\n\n\nSuppose that $c\/b \\in q^{\\mathbb N^*}$. Then the function\n$\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}$ is constant. Hence the\nmatrix $\\breve{P}(1\/b)^{-1}\\breve{P}(z)$ simplifies as follows :\n$$\\breve{P}(1\/b)^{-1}\\breve{P}(z)= \\pmatrice{1}{0}{A\\frac{\\theta_q\n(b z)}{\\theta_q(z)}(1\/z)^{-\\beta}}{1}$$ with $ A\\neq0$. The\nconnection component is equal to $\\pmatrice{1}{0}{\\mathbb C}{1}$\nand the whole Galois group $G$ is equal to\n$\\pmatrice{1}{0}{\\mathbb C}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$.\n\nLast, suppose that $bq\/c \\in q^{\\mathbb N^*}$. Then $A=0$ and the\nfunction $\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}$ is constant, hence\n$G=\\pmatrice{1}{0}{0}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$.\n\\end{proof}\n\n$_{}$\\vskip 10 pt\n\n\\condun{5} $\\underline{a \\in q^{-\\mathbb N}}$.\n\n$_{}$\\vskip 5 pt\n\nIn this case, the twisted connection matrix is upper triangular :\n\n\\begin{eqnarray*}\n\\breve{P}(x) &=&\\pmatrice{\n\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n(-1)^{\\alpha} q^{-\\frac{\\alpha (\\alpha -1)}{2}}}\n{\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{aq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}z^{1-\\gamma}} {0}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c}\nz)}{\\theta_q(z)}(1\/z)^{-\\beta}z^{1-\\gamma}}\n\\end{eqnarray*}\n\n\\begin{theo}\nSuppose that \\condun{5} holds.\nWe have the following trichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $b\/c \\not\\in q^\\mathbb Z$ then\n$G=\\pmatrice{1}{\\mathbb C}{0}{\\mathbb C^*}$;\n\\item[$\\bullet$] if $bq\/c \\in q^{\\mathbb N^*}$ then\n$G=\\pmatrice{1}{\\mathbb C}{0}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$;\n\\item[$\\bullet$] if $c\/b \\in q^{\\mathbb N^*}$ then\n$G=\\pmatrice{1}{0} {0}{\\overline{\\langle w, e^{2 \\pi i\n\\gamma}\\rangle}}$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nWe argue as for theorem \\ref{theo theo}.\n\\end{proof}\n\n$\\bullet$ The cases $\\underline{b \\in q^\\mathbb Z \\text{ or } a\/c \\in q^\\mathbb Z \\text{ or } b\/c \\in q^\\mathbb Z}$ is similar to the case $a \\in q^\\mathbb Z$. We leave the details to the reader.\\\\\n\n\n\\section{Galois groups of the basic hypergeometric equations : logarithmic cases}\n\nWe write $a=uq^\\alpha$, $b=vq^\\beta$ and $c=wq^\\gamma$ with\n$u,v,w\\in\\mathbb{U}$ and $\\alpha,\\beta,\\gamma \\in \\mathbb R$ (we\nchoose a logarithm of $q$).\n\n\n\n\n\\subsection{$c=q$ and $a\/b \\not \\in q^\\mathbb Z$}\n\n\nThe aim of this section is to compute the Galois group of the\nbasic hypergeometric system (\\ref{syst hypergeo}) under the\nassumption : $c=q$ and $a\/b \\not \\in q^\\mathbb Z$.\n\n$_{}$\n\n\\textit{Local fundamental system of solutions at $0$.} We have :\n$$\\qhypermatrice{a}{b}{q}{0}=\\pmatrice{1}{0}{1}{1} \\pmatrice{1}{1}{0}{1} \\pmatrice{1}{0}{1}{1}^{-1}.$$\nConsequently, we are in the non-resonant logarithmic case at $0$.\nWe consider this situation as a degenerate case as $c$ tends to $q$, $c\\neq q$.\n\nMore precisely, we consider the limit as $c$ tends to $q$, with\n$c\\not \\mathbb{E}_q q$, of the following matrix-valued function :\n\n\n\\begin{eqnarray*}\n&&\\fz{a}{b}{c}{z}\\pmatrice{1}{1}{1}{q\/c}^{-1}\\pmatrice{1}{0}{1}{1}\\\\\n&=&\\frac{-c}{c-q} \\pmatrice{(q\/c-1)\\qhyper{a}{b}{c}{z}}{\\qhyper{aq\/c}{bq\/c}{q^2\/c}{z}-\\qhyper{a}{b}{c}{z}}{(q\/c-1)\\qhyper{a}{b}{c}{qz}}{(q\/c) \\qhyper{aq\/c}{bq\/c}{q^2\/c}{qz}-\\qhyper{a}{b}{c}{qz}} \\\\\n\\end{eqnarray*}\n\nUsing the notations : $$\\qhyperc{a}{b}{z} =\n\\frac{d}{dc}_{|c=q}\\left[\\qhyper{a}{b}{c}{z}\\right] \\text{ and }\n\\qhyperabc{a}{b}{z} =\n\\frac{d}{dc}_{|c=q}\\left[\\qhyper{aq\/c}{bq\/c}{q^2\/c}{z}\\right]$$\nthe above limit is equal to :\n\\begin{eqnarray*}\n\\fz{a}{b}{q}{z}&:=&\\pmatrice{\\qhyper{a}{b}{q}{z}}{-q(\\qhyperabc{a}{b}{z} - \\qhyperc{a}{b}{z})}\n{\\qhyper{a}{b}{q}{qz}}{\\qhyper{a}{b}{q}{qz} -q(\\qhyperabc{a}{b}{qz} - \\qhyperc{a}{b}{qz})}. \\\\\n\\end{eqnarray*}\nFrom (\\ref{transfo jauge}) we deduce that $\\fz{a}{b}{q}{z}$\nsatisfies\n$\\fz{a}{b}{q}{qz}\\jz{q}=\\qhypermatrice{a}{b}{c}{z}\\fz{a}{b}{q}{z}$\nwith $\\jz{q}=\\pmatrice{1}{1}{0}{1}$. Hence, this matrix being\ninvertible as a matrix in the field of meromorphic functions, the\nmatrix-valued function\n$\\yz{a}{b}{q}{z}=\\fz{a}{b}{q}{z}e^{(0)}_{\\jz{q}}(z)$ is a\nfundamental system of solutions of the\nbasic hypergeometric equation with $c=q$. Let us recall that $e^{(0)}_{\\jz{q}}(z)=\\pmatrice{1}{\\ell_q(z)}{0}{1}$. \\\\\n\\\\\n\n\\textit{Generators of the local Galois group at $0$.} We have the following generator :\n$$\\pmatrice{1}{1}{0}{1}.$$\\\\\n\n\\textit{Local fundamental system of solutions at $\\infty$.} The\nsituation is as Section \\ref{section generique}. Hence we are in\nthe non-resonant and non-logarithmic case at $\\infty$ and a\nfundamental system of solutions at $\\infty$ of (\\ref{syst\nhypergeo}) as described in Section \\ref{section the basic objects}\nis given by\n$\\yinf{a}{b}{q}{z}=\\finf{a}{b}{q}{z}e^{(\\infty)}_{\\jinf{a}{b}}(z)$\nwith $\\jinf{a}{b} = \\text{diag}(1\/a,1\/b)$ and :\n$$\\finf{a}{b}{q}{z}=\n\\pmatrice{\\qhyper{a}{a}{aq\/b}{\\frac{q^2}{ab}z^{-1}}}{\\qhyper{b}{b}{bq\/a}{\\frac{q^2}{ab}z^{-1}}}\n{\\frac{1}{a} \\qhyper{a}{a}{aq\/b}{\\frac{q}{ab}z^{-1}}}{\\frac{1}{b} \\qhyper{b}{b}{bq\/a}{\\frac{q}{ab}z^{-1}}}.$$\\\\\n\n\n\\textit{Generators of the local Galois group at $\\infty$.} We have two generators :\n$$\\breve{P}(y_0)^{-1}\\pmatrice{e^{2\\pi i\\alpha}}{0}{0}{e^{2\\pi i \\beta}}\\breve{P}(y_0) \\text{ and } \\breve{P}(y_0)^{-1}\\pmatrice{u}{0}{0}{v}\\breve{P}(y_0).$$\\\\\n\n\n\\textit{Connection matrix}. The connection matrix is the limit as $c$ tends to $q$ of :\n\n\\begin{eqnarray*}\n(e^{(\\infty)}_{\\jinf{a}{b}}(z))^{-1}\\pmatrice{\\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (a z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{aq}{c} z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (b z)}{\\theta_q(z)}}\n{\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}\n\\frac{\\theta_q (\\frac{bq}{c} z)}{\\theta_q(z)}}\\pmatrice{1}{1}{1}{q\/c}^{-1}\\pmatrice{1}{0}{1}{1} e^{(0)}_{\\jz{q}}(z)\\\\\n\\end{eqnarray*}\nwhich is equal to :\n\n\\begin{eqnarray*}\nP(z)&:=&(e^{(\\infty)}_{\\jinf{a}{b}}(z))^{-1} \\pmatrice{ u(a,b;q)\n\\frac{\\theta_q (a z)}{\\theta_q(z)}}{\n q(u_c(a,b;q) -v_c(a,b;q))\n\\frac{\\theta_q (a z)}{\\theta_q(z)} +az v(a,b;q) \\frac{\\theta'_q\n(a z)}{\\theta_q(z)}} {w(a,b;q) \\frac{\\theta_q (b\nz)}{\\theta_q(z)}} { q(w_c(a,b;q) -y_c(a,b;q)) \\frac{\\theta_q (b\nz)}{\\theta_q(z)} +bz y(a,b;q) \\frac{\\theta'_q (b\nz)}{\\theta_q(z)}}\ne^{(0)}_{\\jz{q}}(z)\\\\\n\\end{eqnarray*}\nwhere :\n\\begin{eqnarray*}\nu(a,b;c)= \\frac{\\pochamer{b,c\/a}{q}{\\infty}}{\\pochamer{c,b\/a}{q}{\\infty}}; \\ \\ v(a,b;c)=\\frac{\\pochamer{bq\/c,q\/a}{q}{\\infty}}{\\pochamer{q^2\/c,b\/a}{q}{\\infty}};\\\\\nw(a,b;c)=\\frac{\\pochamer{a,c\/b}{q}{\\infty}}{\\pochamer{c,a\/b}{q}{\\infty}};\n\\ \\ y(a,b;c)=\\frac{\\pochamer{aq\/c,q\/b}{q}{\\infty}}{\\pochamer{q^2\/c,a\/b}{q}{\\infty}}.\\\\\n\\end{eqnarray*}\n and where the subscript $c$ means that we take\nthe derivative with respect to the third variable $c$.\\\\\n\n\n\\textit{Twisted connection matrix.}\n\\begin{small} \\begin{eqnarray*}\n\\breve{P}(z)&=&\\pmatrice{(1\/z)^{-\\alpha}}{0}{0}{(1\/z)^{-\\beta}}\n\\pmatrice{ u(a,b;q) \\frac{\\theta_q (a z)}{\\theta_q(z)}}{\n q(u_c(a,b;q) -v_c(a,b;q))\n\\frac{\\theta_q (a z)}{\\theta_q(z)} +az v(a,b;q)\n\\frac{\\theta'_q (a z)}{\\theta_q(z)}} {w(a,b;q) \\frac{\\theta_q\n(b z)}{\\theta_q(z)}} { q(w_c(a,b;q) -y_c(a,b;q))\n\\frac{\\theta_q (b z)}{\\theta_q(z)} +bz y(a,b;q)\n\\frac{\\theta'_q (b z)}{\\theta_q(z)}} \\pmatrice{1}{\\ell_q(z)}{0}{1}\n\\end{eqnarray*}\\end{small}\n\n$_{}$\\vskip 10 pt\n\nWe need to consider different cases.\n\n$_{}$\\vskip 10 pt\n\n\\conddeux{1} $\\underline{a \\not\\in q^\\mathbb Z \\text{ and } b\\not \\in\nq^\\mathbb Z}$.\n\n$_{}$\\vskip 5 pt\n\nSubject to this condition, the complex numbers $u(a,b;q)$,\n$v(a,b;q)$, $w(a,b;q)$ and $y(a,b;q)$ are non-zero.\n\n\\begin{prop} \\label{blabla}\nIf \\conddeux{1} holds then the natural action of $G^I$ on $\\mathbb C^2$ is irreducible.\n\\end{prop}\n\n\\begin{proof} Assume, at the contrary, that the action of $G^I$ is\nreducible and let $L$ be an invariant line.\n\nLet us fist remark that $L \\neq \\mathbb C \\vect{1}{0}$ (in\nparticular, $G^I$ does not consist of scalar matrices). Indeed, if not,\n$\\mathbb C \\vect{1}{0}$ would be stabilized by the connection\ncomponent and the line spanned by $\\breve{P}(z) \\vect{1}{0}$\nwould be independent of $z\\in\\Omega$ : this is clearly false.\n\nThe group $G^I$ being normalized by $\\pmatrice{1}{1}{0}{1}$\n(since $G^I$ is a normal subgroup of $G$), the\nlines $\\pmatrice{1}{1}{0}{1}^n L$ are also invariant by the action\nof $G^I$. These lines being distinct (since $L\\neq \\mathbb C\n\\vect{1}{0}$) we conclude that $G^I$ consists of scalar matrices and\nwe get a contradiction. This proves that $G^I$ acts\nirreducibly.\n\\end{proof}\n\nAs a consequence we have the following theorem.\n\n\\begin{theo}\nSuppose that \\conddeux{1} holds. Then we have the following dichotomy :\n\\begin{itemize}\n\\item[$\\bullet$] if $ab \\not \\in q^\\mathbb Z$ then $G=\\text{Gl}_2(\\mathbb C)$;\n\\item[$\\bullet$] if $ab \\in q^\\mathbb Z$ then $G= \\text{Sl}_2(\\mathbb C)$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nUsing the irreducibility of the natural action of $G^I$ and\narguing as for the proof of theorem \\ref{theo un}, we obtain the\nequality $G^{I,der}=\\text{Sl}_2(\\mathbb C)$. From the formula (\\ref{det})\nwe deduce that the determinant of the twisted connection matrices\nwhen $c=q$ is equal to the limit as $c$ tends to $q$ of\n$\\frac{-1}{1\/a-1\/b}(1\/z)^{-(\\alpha+\\beta)} z^{1-\\gamma}\n\\frac{\\theta_q (\\frac{abq}{c} z)}{\\theta_q(z)}$ \\textit{i.e.}\n$\\frac{-1}{1\/a-1\/b}(1\/z)^{-(\\alpha+\\beta)} z^{1-\\gamma}\n\\frac{\\theta_q (ab z)}{\\theta_q(z)}.$\n\nIf $ab \\not \\in q^\\mathbb Z$ then this determinant is a\nnon-constant holomorphic function and consequently\n$G=\\text{Gl}_2(\\mathbb C)$.\n\nIf $ab \\in q^\\mathbb Z$ then this determinant does not depend on\n$z$. This implies that the connection component of the Galois\ngroup is a sub-group of $\\text{Sl}_2(\\mathbb C)$. Furthermore, $ab \\in\nq^\\mathbb Z$ entails that $uv =1$ and $\\alpha+\\beta \\in\n\\mathbb Z$, that is, $e^{2\\pi i (\\alpha+\\beta)}=1$. Consequently,\nthe local Galois groups are subgroups of $\\text{Sl}_2(\\mathbb C)$ and the\nglobal Galois group $G$ is therefore a subgroup of\n$\\text{Sl}_2(\\mathbb C)$.\n\\end{proof}\n\n\n\n$_{}$\\vskip 10 pt\n\n\\conddeux{2} $\\underline{b\\in q^{\\mathbb N^*}}$.\n\n$_{}$\\vskip 5 pt\n\nThen the twisted\nconnection matrix simplifies as follows :\n\\begin{small}\\begin{eqnarray*}\n\\breve{P}(z)&=& \\pmatrice{ u(a,b;q) \\frac{\\theta_q (a\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}} {q(u_c(a,b;q) -v_c(a,b;q))\n\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha} +az v(a,b;q)\n\\frac{\\theta'_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}} {0} {\nq(w_c(a,b;q) -y_c(a,b;q)) (-1)^\\beta\nq^{-\\frac{\\beta(\\beta-1)}{2}}}\n\\pmatrice{1}{\\ell_q(z)}{0}{1}\\\\\n\\end{eqnarray*}\\end{small}\n\n\\begin{theo}\\label{theou}\nSuppose that \\conddeux{2} holds.\nThen we have $G=\\pmatrice{\\mathbb C^*}{\\mathbb C}{0}{1}$.\n\\end{theo}\n\\begin{proof}\nFix a point $y_0 \\in \\Omega$ such that $\\breve{P}(y_0)$ is of the\nform $\\pmatrice{A}{B}{0}{C}$ with $A,C \\neq 0$. There exists a\nconstant $D\\in \\mathbb C^*$ such that :\n$$\\breve{P}(y_0)^{-1}\\breve{P}(z)= \\pmatrice{D\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}{*}{0}{1}.$$\nSince $G^I$ is normalized by $\\pmatrice{1}{1}{0}{1}$\n(remember that $G^I$ is a normal subgroup of $G$), it contains, for all $n\\in \\mathbb Z$, the matrix :\n$$\\pmatrice{D\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}{*+n(D\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}-1)}{0}{1}.$$\nBecause $a\\not \\in q^\\mathbb Z$, the function $D\\frac{\\theta_q (a\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}-1$ is not identically equal to\nzero over $\\mathbb C^*$ and therefore $G^I$ contains, for all $z\n\\in \\Omega$ :\n$$\\pmatrice{D\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}{\\mathbb C}{0}{1}.$$\nIn particular, $\\pmatrice{D\\frac{\\theta_q (a\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}}{0}{0}{1}$ belongs to $G^I$, so that\n$\\pmatrice{\\mathbb C^*}{0}{0}{1}$ is a subgroup of $G^I$ and\n$\\pmatrice{\\mathbb C^*}{\\mathbb C}{0}{1} \\subset G$. The converse\ninclusion is clear.\n\\end{proof}\n\n$_{}$\\vskip 10 pt\n\n\\conddeux{3} $\\underline{b\\in q^{-\\mathbb N}}$.\n\n$_{}$\\vskip 5 pt\n\nUsing the identity :\n$$bz\\frac{\\theta_q'(bz)}{\\theta_q(z)}=(-\\beta-\\ell_q(z))(-1)^\\beta\nq^{-\\frac{\\beta(\\beta-1)}{2}}$$ we see that, in this case, the\ntwisted connection matrix takes the form :\n\n\\begin{eqnarray*}\n\\breve{P}(z\n&=&\\pmatrice{0} {q(u_c(a,b;q) -v_c(a,b;q)) \\frac{\\theta_q (a\nz)}{\\theta_q(z)}(1\/z)^{-\\alpha}} {w(a,b,q) (-1)^\\beta\nq^{-\\frac{\\beta(\\beta-1)}{2}}} { q(w_c(a,b;q)\n-y_c(a,b;q)-\\beta\/q) (-1)^\\beta q^{-\\frac{\\beta(\\beta-1)}{2}}}\\\\\n\\end{eqnarray*}\n\n\\begin{theo}\nSuppose that \\conddeux{3} holds. Then we have $G=\\pmatrice{1}{\\mathbb C}{0}{\\mathbb C^*}.$\n\\end{theo}\n\n\\begin{proof}\nFix a base point $y_0 \\in \\Omega$. There exist three constants\n$C,C',C''\\in \\mathbb C$ with $C\\neq 0$, such that the following\nidentity holds for all $z\\in \\Omega$ :\n$$\\breve{P}(y_0)^{-1}\\breve{P}(z)=\\pmatrice{1}{C'\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}+C''}\n{0}{C\\frac{\\theta_q (a z)}{\\theta_q(z)}(1\/z)^{-\\alpha}}.$$ The proof is similar to the proof of Theorem \\ref{theou}.\n\\end{proof}\n\n\nThe remaining case $a\\in q^\\mathbb Z$ is similar to\n\\conddeux{3}.\n\nThe case $a=b$ and $c\\not \\in q^\\mathbb Z$ is similar to the case treated in this section.\n\n\n\\subsection{$a=b$ and $c=q$}\n\nThe aim of this section is to compute the Galois group of the\nbasic hypergeometric system (\\ref{syst hypergeo}) under the\nassumption : $a=b$ and $c=q$.\n\n$_{}$\n\n\\textit{Local fundamental system of solutions at $0$.} The\nsituation is the same\nas in the case $c=q$ and $a\/b \\not\\in q^\\mathbb Z$.\\\\\n\n\\textit{Generator of the local Galois group at $0$}. We have the following generator :\n$$\\pmatrice{1}{1}{0}{1}.$$\\\\\n\n\\textit{Local fundamental system of solutions at $\\infty$.} We\nhave :\n$$\\qhypermatrice{a}{a}{q}{z}=\\pmatrice{1}{0}{1\/a}{1} \\pmatrice{1\/a}{1}{0}{1\/a} \\pmatrice{1}{0}{1\/a}{1}^{-1}.$$\nConsequently, we are in the non-resonant logarithmic case at\n$\\infty$. We consider the case $a=b$ and $c=q$ as a degenerate\ncase of the situation $c=q$ as $a$ tends to $b$, $a\/b \\neq 1$.\n\n\nWe consider the following matrix valued function :\n$$\\finf{a}{b}{q}{z}\\pmatrice{1}{1}{1\/a}{1\/b}^{-1}\\pmatrice{1}{0}{1\/a}{1}.$$\nA straightforward calculation, which we omit here because it is\nlong although easy, shows that this matrix-valued function does admit a limit as $a$ tends to $b$ that we denote $\\finf{a}{a}{q}{z}$. A fundamental\nsystem of solutions at $\\infty$ of (\\ref{syst hypergeo}) as\ndescribed in section \\ref{section the basic objects} is given by\n$\\yinf{a}{a}{q}{z}=\\finf{a}{a}{q}{z}e^{(\\infty)}_{\\jinf{a}{a}}(z)$\nwith $\\jinf{a}{a} = \\pmatrice{1\/a}{1}{0}{1\/a}$.\\\\\n\\\\\n\n\\textit{Generator of the local Galois group at $\\infty$}. We have the following generators :\n$$\\pmatrice{u}{0}{0}{u}, \\ \\ \\pmatrice{e^{2\\pi i \\alpha}}{0}{0}{e^{2\\pi i \\alpha}} \\text{ and } \\breve{P}(y_0)^{-1}\\pmatrice{1}{a}{0}{1}\\breve{P}(y_0).$$\\\\\n\n\\textit{Birkhoff matrix}. The Birkhoff matrix is equal to\n$(e^{(\\infty)}_{\\jinf{a}{a}}(z))^{-1}Qe^{(0)}_{\\jz{q}}(z)$ where\n$Q$ is the limit as $a$ tends to $b$ of :\n\n\\begin{eqnarray*}\n&&\\pmatrice{1}{0}{1\/a}{1}^{-1}\\pmatrice{1}{1}{1\/a}{1\/b}\\pmatrice{\nu(a,b;q) \\frac{\\theta_q (a z)}{\\theta_q(z)}}{\n q(u_c(a,b;q) -v_c(a,b;q))\n\\frac{\\theta_q (a z)}{\\theta_q(z)} +az v(a,b;q)\n\\frac{\\theta'_q (a z)}{\\theta_q(z)}} {w(a,b;q) \\frac{\\theta_q\n(b z)}{\\theta_q(z)}} { q(w_c(a,b;q) -y_c(a,b;q))\n\\frac{\\theta_q (b z)}{\\theta_q(z)} +bz y(a,b;q)\n\\frac{\\theta'_q (b z)}{\\theta_q(z)}}.\n\\end{eqnarray*}\nIt has the following form :\n\n\n\\begin{eqnarray*}\n&& \\pmatrice{ C \\frac{\\theta_q(az)}{\\theta_q(z)} +az\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2} \\frac{\\theta_q'(a\nz)}{\\theta_q(z)}} {*} {-(1\/a)\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\\frac{\\theta_q(az)}{\\theta_q(z)}}\n{C'\\frac{\\theta_q(az)}{\\theta_q(z)}-z\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\n\\frac{\\theta_q'(a z)}{\\theta_q(z)}}\n\\end{eqnarray*}\nwhere $*$ denotes some meromorphic function.\\\\\n\n\\textit{Twisted Birkhoff matrix}.\n\n$$\n (1\/z)^{-\\alpha} \\pmatrice{1}{-a\\ell_q(z)}{0}{1}\n\\pmatrice{C \\frac{\\theta_q(az)}{\\theta_q(z)}+az\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\n\\frac{\\theta_q'(a z)}{\\theta_q(z)}} {*} {-(1\/a)\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\\frac{\\theta_q(az)}{\\theta_q(z)}}\n{C'\\frac{\\theta_q(az)}{\\theta_q(z)}-z\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\n\\frac{\\theta_q'(a z)}{\\theta_q(z)}}\n \\pmatrice{1}{\\ell_q(z)}{0}{1}.\n$$\n\n$_{}$\\vskip 10 pt\n\nWe need to consider different cases.\n\n$_{}$\\vskip 10 pt\n\n\\condtrois{1} $\\underline{a \\not \\in q^\\mathbb Z}$.\n\n$_{}$\\vskip 5 pt\n\n\\begin{prop}\nSuppose that \\condtrois{1} holds. Then the natural action of $G^I$ on $\\mathbb C^2$ is irreducible.\n\\end{prop}\n\\begin{proof}\nRemark that $\\mathbb C \\vect{1}{0}$ is not an invariant line.\nIndeed, if not, this line would be invariant by the action of the\nconnection component, hence the line spanned by $\\breve{P}(z)\n\\vect{1}{0}$ would be independent of $z\\in\\Omega$. Considering the\nratio of the coordinates of this line, this would imply the\nexistence of some constant $A \\in \\mathbb C$ such that the\nfollowing functional equation holds on $\\mathbb C^*$ :\n$$C \\frac{\\theta_q(az)}{\\theta_q(z)} +az\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2} \\frac{\\theta_q'(a\nz)}{\\theta_q(z)}+\n\\frac{\\theta_q(a)}{\\pochamer{q}{q}{\\infty}^2}\\frac{\\theta_q(az)}{\\theta_q(z)}\n\\ell_q(z) = A \\frac{\\theta_q(az)}{\\theta_q(z)}.$$ The fact that\n$\\theta_q(az)$ vanishes exactly to the order one at $z=1\/a$,\nyields a contradiction.\n\nThe end of the proof is similar to the proof of Proposition\n\\ref{blabla}.\n\\end{proof}\n\n\\begin{theo}\nIf \\condtrois{1} holds then we have the following dichotomy :\n\\begin{itemize}\n \\item[$\\bullet$] if $a^2 \\not \\in q^\\mathbb Z$ then\n $G=\\text{Gl}_2(\\mathbb C)$;\n \\item[$\\bullet$] if $a^2 \\in q^\\mathbb Z$ then\n $G=\\text{Sl}_2(\\mathbb C)$.\n\\end{itemize}\n\\end{theo}\n\n\\begin{proof}\nThe proof follows the same line as that of theorem \\ref{theo un}.\n\\end{proof}\n\n$_{}$\\vskip 10 pt\n\n\\condtrois{2} $\\underline{a \\in q^\\mathbb Z}$.\n\n$_{}$\\vskip 5 pt\n\nUnder this condition, the connection matrix simplifies as follows,\nfor some constants $C,C'\\in \\mathbb C$ :\n\\begin{eqnarray*}\n&&\n\\pmatrice{ C } {*}\n{0}{C'}.\\\\\n\\end{eqnarray*}\n\n\n\\begin{theo}\nSuppose that \\condtrois{2} holds. Then we have : $G = \\pmatrice{1}{\\mathbb C}{0}{1}$.\n\\end{theo}\n\\begin{proof}\nThe local Galois group at $0$ is generated by\n$\\pmatrice{1}{1}{0}{1}$, hence $G$ contains\n$\\pmatrice{1}{\\mathbb C}{0}{1}$.\n\nSince the twisted connection matrix is upper triangular with\nconstant diagonal entries, the connection component is a subgroup of\n$\\pmatrice{1}{\\mathbb C}{0}{1}$. The generators of the local Galois\ngroup at $0$ and at $\\infty$ also lie in\n$\\pmatrice{1}{\\mathbb C}{0}{1}$. Therefore, $G$ is a subgroup of\n$\\pmatrice{1}{\\mathbb C}{0}{1}$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}