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+{"text":"\\section{Introduction}\nAlternating direction method of multipliers (ADMM \\cite{gabay1976dual,boyd2011distributed}) is a popular optimization tool\nfor solving the composite and constrained problems in machine learning.\nIn particular, ADMM can efficiently optimize some problems with complicated structure regularization\nsuch as the graph-guided fused lasso \\cite{kim2009multivariate},\nwhich is too complicated for the other popular optimization methods\nsuch as proximal gradient methods \\cite{beck2009fast}.\nFor the large-scale optimization,\nthe stochastic ADMM method \\cite{ouyang2013stochastic} has been proposed.\nRecently, some faster stochastic ADMM methods \\cite{suzuki2014stochastic,zheng2016fast}\nhave been proposed by using the variance reduced (VR) techniques such as the SVRG \\cite{johnson2013accelerating}.\nIn fact, ADMM is also highly successful in solving various nonconvex problems\nsuch as training deep neural networks \\cite{Taylor2016Training}.\nThus, some fast nonconvex stochastic ADMM methods have been developed in \\cite{huang2016stochastic,Huang2019fasterst}.\n\nCurrently, most of the ADMM methods\nneed to  compute the gradients\nof objective functions over each iteration.\nHowever, in many machine learning problems,\nthe explicit expression of gradient for objective function\nis difficult or infeasible to obtain.\nFor example, in black-box situations, only prediction results (\\emph{i.e.}, function\nvalues) are provided \\cite{chen2017zoo,liu2018zeroth}.\nIn bandit settings \\cite{agarwal2010optimal},\nplayer only receives the partial feedback in terms of loss function values,\nso it is impossible to obtain expressive gradient of the loss function.\nClearly, the classic optimization methods, based on the first-order gradient or second-order information, are not competent to these problems.\nRecently, the zeroth-order optimization methods \\cite{duchi2015optimal,Nesterov2017RandomGM}\nare developed by only using the function values in the optimization.\n\\begin{table*}\n  \\centering\n  \\begin{tabular}{c|c|c|c|c}\n  \\hline\n \n \\textbf{Algorithm} & \\textbf{Reference} & \\textbf{Gradient Estimator} &  \\textbf{Problem}  & \\textbf{Convergence Rate} \\\\ \\hline\n  ZOO-ADMM & \\cite{liu2018admm} &GauSGE &  C(S) + C(NS) & $O(\\sqrt{1\/T})$  \\\\ \\hline\n  ZO-GADM & \\cite{gao2018information} & UniSGE & C(S) + C(NS)&  $O(\\sqrt{1\/T})$  \\\\ \\hline\n  RSPGF  &\\cite{ghadimi2016mini} & GauSGE &NC(S) + C(NS)  & $O(\\sqrt{1\/T})$ \\\\ \\hline\n  ZO-ProxSVRG & \\multirow{2}*{\\cite{Huang2019faster}} & \\multirow{2}*{CooSGE} & \\multirow{2}*{NC(S) + C(NS)} & \\multirow{2}*{$O(1\/T)$} \\\\\n  ZO-ProxSAGA &  & & &   \\\\ \\hline\n  ZO-SVRG-ADMM & \\multirow{2}*{Ours} & \\multirow{2}*{CooSGE} & \\multirow{2}*{NC(S) + C(mNS)} & \\multirow{2}*{$O(1\/T)$} \\\\\n  ZO-SAGA-ADMM &  & & &   \\\\ \\hline\n  \\end{tabular}\n  \\caption{  Convergence properties comparison of the zeroth-order ADMM algorithms and other ones.\n  C, NC, S, NS and mNS are the abbreviations of convex,\n    non-convex, smooth, non-smooth and the sum of multiple non-smooth functions, respectively.\n    $T$ is the whole iteration number.\n    Gaussian Smoothing Gradient Estimator (GauSGE),\n    Uniform Smoothing Gradient Estimator (UniSGE) and Coordinate Smoothing Gradient Estimator (CooSGE). }\\label{tab:1}\n\\end{table*}\n\nIn the paper, we focus on using the zeroth-order methods to solve the following nonconvex nonsmooth problem:\n\\begin{align} \\label{eq:1}\n \\min_{x,\\{y_j\\}_{j=1}^k} & F(x,y_{[k]}) := \\frac{1}{n}\\sum_{i=1}^n f_i(x) + \\sum_{j=1}^{k}\\psi_j(y_j)  \\\\\n \\mbox{s.t.} \\ & Ax + \\sum_{j=1}^kB_jy_j =c, \\nonumber\n\\end{align}\nwhere $A\\in \\mathbb{R}^{p\\times d}$, $B_j\\in \\mathbb{R}^{p\\times q}$ for all $j\\in [k],\\ k \\geq 1$,\n$f(x)=\\frac{1}{n}\\sum_{i=1}^n f_i(x): \\mathbb{R}^d\\rightarrow \\mathbb{R}$\nis a \\emph{nonconvex} and black-box function, and each $\\psi_j(y_j): \\mathbb{R}^q\\rightarrow \\mathbb{R}$ is a convex and\n\\emph{nonsmooth} function.\nIn machine learning, function $f(x)$ can be used for the empirical loss,\n$\\sum_{j=1}^{k}\\psi_j(y_j)$ for multiple structure penalties (\\emph{e.g.}, sparse + group sparse), and the constraint for encoding\nthe structure pattern of model parameters such as graph structure.\nDue to the flexibility in splitting the objective function into loss $f(x)$ and each penalty $\\psi_j(y_j)$,\nADMM is an efficient method to solve the obove problem. However,\nin the problem \\eqref{eq:1}, we only access the objective values\nrather than the whole explicit function $F(x,y_{[k]})$,\nthus the classic ADMM methods are unsuitable for the problem \\eqref{eq:1}.\n\nRecently, \\cite{gao2018information,liu2018admm} proposed the zeroth-order stochastic ADMM methods,\nwhich only use the objective values to optimize. However,\nthese zeroth-order ADMM-based methods build on the convexity of objective function. Clearly, these methods\nare limited in many nonconvex problems such as adversarial attack on black-box deep neural network (DNN).\nAt the same time, due to that the problem \\eqref{eq:1} includes multiple nonsmooth regularization functions and an equality constraint,\nthe existing zeroth-order algorithms \\cite{liu2018zeroth,Huang2019faster}\nare not suitable for solving this problem.\n\nIn the paper, thus, we propose a class of\nfast zeroth-order stochastic ADMM methods (\\emph{i.e.}, ZO-SVRG-ADMM and ZO-SAGA-ADMM) to solve the problem \\eqref{eq:1} based on the coordinate\nsmoothing gradient estimator \\cite{liu2018zeroth}.\nIn particular, the ZO-SVRG-ADMM and ZO-SAGA-ADMM methods build on\nthe SVRG \\cite{johnson2013accelerating} and SAGA \\cite{defazio2014saga}, respectively.\nMoreover, we study the convergence properties of the proposed methods.\nTable \\ref{tab:1} shows the convergence properties\nof the proposed methods and other related ones.\n\\subsection{Challenges and Contributions}\nAlthough both SVRG and SAGA show good performances in the first-order and second-order methods,\napplying these techniques to the nonconvex zeroth-order ADMM method\nis \\emph{not trivial}. There exists at least two main \\textbf{challenges}:\n\\begin{itemize}\n \\item Due to failure of the Fej\\'{e}r monotonicity of iteration, the convergence analysis of the nonconvex ADMM is generally quite difficult \\cite{wang2015convergence}.\n       With using the inexact zeroth-order estimated gradient, this difficulty becomes greater in the nonconvex ADMM methods.\n \\item To guarantee convergence of our zeroth-order ADMM methods, we need to design a new effective \\emph{Lyapunov} function, which can not follow\n       the existing nonconvex (stochastic) ADMM methods \\cite{jiang2019structured,huang2016stochastic}.\n\\end{itemize}\nThus, we carefully establish the \\emph{Lyapunov} functions in the following theoretical analysis to\nensure convergence of the proposed methods.\nIn summary, our major \\textbf{contributions} are given below:\n\\begin{itemize}\n\\item[1)] We propose a class of fast zeroth-order stochastic ADMM methods (\\emph{i.e.}, ZO-SVRG-ADMM and ZO-SAGA-ADMM) to solve the problem \\eqref{eq:1}.\n\\item[2)] We prove that both the ZO-SVRG-ADMM and ZO-SAGA-ADMM have convergence rate of $O(\\frac{1}{T})$ for nonconvex nonsmooth optimization.\n          In particular, our methods not only reach the existing best convergence rate $O(\\frac{1}{T})$ for the nonconvex optimization,\n          but also are able to effectively solve many machine learning problems with multiple complex regularized penalties.\n\\item[3)] Extensive experiments conducted on black-box classification and structured adversarial attack on black-box DNNs\n          validate efficiency of the proposed algorithms.\n\\end{itemize}\n\\section{Related Works}\nZeroth-order (gradient-free) optimization is a powerful optimization tool for solving many\nmachine learning problems, where the gradient\nof objective function is not available or computationally prohibitive.\nRecently, the zeroth-order optimization methods are widely applied and studied.\nFor example, zeroth-order optimization methods have been applied to bandit feedback analysis \\cite{agarwal2010optimal} and\nblack-box attacks on DNNs \\cite{chen2017zoo,liu2018zeroth}.\n\\cite{Nesterov2017RandomGM} have proposed several random zeroth-order\nmethods based on the Gaussian smoothing gradient estimator.\nTo deal with the nonsmooth regularization, \\cite{gao2018information,liu2018admm} have proposed\nthe zeroth-order online\/stochastic ADMM-based methods.\n\nSo far, the above algorithms mainly build on the convexity of problems.\nIn fact, the zeroth-order methods are also highly successful in solving various nonconvex problems such as\nadversarial attack to black-box DNNs \\cite{liu2018zeroth}.\nThus, \\cite{ghadimi2013stochastic,liu2018zeroth,Gu2018faster}\nhave begun to study the zeroth-order stochastic methods for the nonconvex optimization.\nTo deal with the nonsmooth regularization, \\cite{ghadimi2016mini,Huang2019faster} have proposed\nsome non-convex zeroth-order proximal stochastic gradient methods.\nHowever, these methods still are not well competent to some complex machine learning problems\nsuch as a task of structured adversarial attack to the black-box DNNs, which is described in the following experiment.\n\\subsection{Notations}\nLet $y_{[k]}= \\{y_1,\\cdots,y_k\\}$ and $y_{[j:k]}= \\{y_{j},\\cdots,y_k\\}$ for $j\\in[k]$.\nGiven a positive definite matrix $G$, $\\|x\\|^2_G = x^TGx$;\n$\\sigma_{\\max}(G)$ and $\\sigma_{\\min}(G)$\ndenote the largest and smallest eigenvalues of $G$, respectively,\nand $\\kappa_G = \\frac{\\sigma_{\\max}(G)}{\\sigma_{\\min}(G)}$.\n$\\sigma^A_{\\max}$ and $\\sigma^A_{\\min}$ denote the largest and smallest eigenvalues of matrix $A^TA$.\n\\section{Preliminaries}\nIn the section, we begin with restating a standard $\\epsilon$-approximate stationary point of the problem \\eqref{eq:1},\nas in \\cite{jiang2019structured,Huang2019fasterst}.\n\\begin{definition} \\label{def:1}\nGiven $\\epsilon>0$, the point $(x^*,y_{[k]}^*,\\lambda^*)$ is said to be an\n$\\epsilon$-approximate stationary point of the problems \\eqref{eq:1},\nif it holds that\n\\begin{align}\n \\mathbb{E}\\big[ \\mbox{dist}(0,\\partial L(x^*,y_{[k]}^*,\\lambda^*))^2 \\big] \\leq \\epsilon,\n\\end{align}\nwhere $L(x,y_{[k]},\\lambda)=f(x) + \\sum_{j=1}^k\\psi_j(y_j) - \\langle \\lambda, Ax+\\sum_{j=1}^kB_jy_j-c\\rangle$,\n\\begin{align}\n   \\partial L(x,y_{[k]},\\lambda) = \\left [ \\begin{matrix}\n     \\nabla_x L(x,y_{[k]},\\lambda) \\\\\n     \\partial_{y_1} L(x,y_{[k]},\\lambda) \\\\\n     \\cdots \\\\\n     \\partial_{y_k} L(x,y_{[k]},\\lambda) \\\\\n     -Ax - \\sum_{j=1}^kB_jy_j+c\n \\end{matrix}\n \\right ], \\nonumber\n\\end{align}\n$\\mbox{dist}(0,\\partial L)=\\inf_{L'\\in \\partial L} \\|0-L'\\|.$\n\\end{definition}\n\nNext, we make some mild assumptions regarding problem \\eqref{eq:1} as follows:\n\\begin{assumption}\nEach function $f_i(x)$ is $L$-smooth for $\\forall i \\in \\{1,2,\\cdots,n\\}$ such that\n\\begin{align}\n\\|\\nabla f_i(x)-\\nabla f_i(y)\\| \\leq L \\|x - y\\|, \\ \\forall x,y \\in \\mathbb{R}^d, \\nonumber\n\\end{align}\nwhich is equivalent to\n\\begin{align}\nf_i(x) \\leq f_i(y) + \\nabla f_i(y)^T(x-y) + \\frac{L}{2}\\|x-y\\|^2.  \\nonumber\n\\end{align}\n\\end{assumption}\n\\begin{assumption}\nFull gradient of loss function $f(x)$ is bounded, i.e., there exists a constant $\\delta >0$ such that for all $x$,\nit follows that $\\|\\nabla f(x)\\|^2 \\leq \\delta^2$.\n\\end{assumption}\n\\begin{assumption}\n$f(x) $ and $\\psi_j(y_j)$ for all $j\\in [k]$ are all lower bounded, and denote\n$f^* = \\inf_x f(x)$ and $\\psi_j^* = \\inf_y \\psi_j(y)$ for $j\\in [k]$.\n\\end{assumption}\n\\begin{assumption}\n$A$ is a full row or column rank matrix.\n\\end{assumption}\nAssumption 1 has been commonly used in the convergence analysis of\nnonconvex algorithms \\cite{ghadimi2016mini}.\nAssumption 2 is widely used for\nstochastic gradient-based and ADMM-type methods \\cite{boyd2011distributed}.\nAssumptions 3 and 4 are usually used in the convergence analysis of ADMM methods \\cite{jiang2019structured,huang2016stochastic,Huang2019fasterst}.\nWithout loss of generality, we will use the full column rank of matrix $A$ in the rest of this paper.\n\\section{ Fast Zeroth-Order Stochastic ADMMs }\n\\begin{algorithm}[htb]\n   \\caption{ Nonconvex ZO-SVRG-ADMM Algorithm }\n   \\label{alg:1}\n\\begin{algorithmic}[1]\n   \\STATE {\\bfseries Input:} $b$, $m$, $T$, $S=\\lceil T\/m\\rceil$, $\\eta>0$ and $\\rho>0$;\n   \\STATE {\\bfseries Initialize:} $x_0^1$, $y^{0,1}_j$ for $j\\in [k]$ and $\\lambda_0^1$;\n   \\FOR {$s=1,2,\\cdots,S$}\n   \\STATE{} $\\tilde{x}^{s+1}=x_0^{s+1}$, $\\hat{\\nabla} f(\\tilde{x}^{s})=\\frac{1}{n}\\sum_{i=1}^n\\hat{\\nabla} f_i(\\tilde{x}^{s})$;\n   \\FOR {$t=0,1,\\cdots,m-1$}\n   \\STATE{} Uniformly randomly pick a mini-batch $\\mathcal{I}_t$ (with replacement) from $\\{1,2,\\cdots,n\\}$, and $|\\mathcal{I}_t|=b$ ;\n   \\STATE{}  Using \\eqref{eq:5} to estimate stochastic gradient\n             $\\hat{g}_{t}^{s} = \\hat{\\nabla} f_{\\mathcal{I}_t}(x_{t}^{s})-\\hat{\\nabla} f_{\\mathcal{I}_t}(\\tilde{x}^s)+\\hat{\\nabla} f(\\tilde{x}^s)$;\n   \\STATE{}  $y_j^{s,t+1} \\!= \\! \\arg\\min_{y_j} \\! \\big\\{ \\mathcal {L}_{\\rho}(x_t^s,y_{[j-1]}^{s,t+1},y_j,y_{[j+1:k]}^{s,t},\\lambda_t^s)$\n             $ \\quad + \\frac{1}{2}\\|y_j-y_j^{s,t}\\|^2_{H_j} \\big\\} $, for all $j\\in [k]$;\n   \\STATE{}  $x^s_{t+1}= \\arg\\min_x \\hat{\\mathcal {L}}_{\\rho}\\big( x,y_{[k]}^{s,t+1},\\lambda^s_t, \\hat{g}_{t}^{s} \\big)$;\n   \\STATE{}  $\\lambda^s_{t+1} = \\lambda^s_t-\\rho(Ax^s_{t+1} + \\sum_{j=1}^kB_jy_j^{s,t+1}-c)$;\n   \\ENDFOR\n   \\STATE{} $x_0^{s+1}=x_{m}^{s}$, $y_j^{s+1,0}=y_j^{s,m}$ for $j\\in [k]$, $\\lambda_0^{s+1}=\\lambda_{m}^{s}$;\n   \\ENDFOR\n   \\STATE {\\bfseries Output:} $\\{x,y_{[k]},\\lambda\\}$ chosen at random uniformly from $\\{(x_{t}^s,y_{[k]}^{s,t},\\lambda_t^s)_{t=1}^{m}\\}_{s=1}^S$.\n\\end{algorithmic}\n\\end{algorithm}\n\nIn this section, we propose a class of zeroth-order stochastic ADMM methods to solve\nthe problem \\eqref{eq:1}. First, we define an augmented\nLagrangian function of the problem \\eqref{eq:1}:\n\\begin{align}\n \\mathcal {L}_{\\rho}(x,y_{[k]},\\lambda) & \\!= f(x) + \\sum_{j=1}^k\\psi_j(y_j) \\!-\\! \\langle\\lambda, Ax \\!+\\! \\sum_{j=1}^kB_jy_j-c\\rangle \\nonumber \\\\\n &  + \\frac{\\rho}{2} \\|Ax + \\sum_{j=1}^kB_jy_j - c\\|^2,\n\\end{align}\nwhere $\\lambda\\in \\mathbb{R}^{p}$ and $\\rho >0$ denotes the dual variable and penalty parameter, respectively.\n\n\\begin{algorithm}[htb]\n   \\caption{ Nonconvex ZO-SAGA-ADMM Algorithm }\n   \\label{alg:2}\n\\begin{algorithmic}[1]\n   \\STATE {\\bfseries Input:} $b$, $T$, $\\eta>0$ and $\\rho>0$;\n   \\STATE {\\bfseries Initialize:} $z_i^0=x_0$ for $i\\in \\{1,2,\\cdots,n\\}$, $\\hat{\\phi}_0=\\frac{1}{n}\\sum_{i=1}^n\\nabla f_i(z^0_i)$, $y^{0}_j$ for $j\\in [k]$ and $\\lambda_0$;\n   \\FOR {$t=0,1,\\cdots,T-1$}\n   \\STATE{} Uniformly randomly pick a mini-batch $\\mathcal{I}_t$ (with replacement) from $\\{1,2,\\cdots,n\\}$, and $|\\mathcal{I}_t|=b$ ;\n   \\STATE{} Using \\eqref{eq:5} to estimate stochastic gradient $\\hat{g}_{t} \\!=\\! \\frac{1}{b}\\sum_{i_t\\in \\mathcal{I}_t}\n             \\big(\\hat{\\nabla} f_{i_t}(x_{t})\\!-\\!\\hat{\\nabla} f_{i_t}(z^t_{i_t}) \\big)\\!+\\!\\hat{\\phi}_t$\n            with $\\hat{\\phi}_t\\!=\\!\\frac{1}{n}\\sum_{i=1}^n\\hat{\\nabla} f_i(z^t_i)$;\n   \\STATE{}  $y_j^{t+1} = \\arg\\min_{y_j} \\big\\{ \\mathcal {L}_{\\rho}(x_t,y_{[j-1]}^{t+1},y_j,y_{[j+1:k]}^{t},\\lambda_t)\n              + \\frac{1}{2}\\|y_j-y_j^t\\|^2_{H_j} \\big\\}$, for all $j\\in [k]$;\n   \\STATE{}  $x_{t+1}= \\arg\\min_x \\hat{\\mathcal {L}}_{\\rho}\\big( x,y_{[k]}^{t+1},\\lambda_t, \\hat{g}_{t} \\big)$;\n   \\STATE{}  $\\lambda_{t+1} = \\lambda_t-\\rho(Ax_{t+1} + \\sum_{j=1}^kB_jy_j^{t+1}-c)$;\n   \\STATE{} $z^{t+1}_{i_t}= x_{t}$ for $i_t \\in \\mathcal{I}_t$ and $z_{i_t}^{t+1}=z^t_{i_t}$ for $i_t \\not\\in \\mathcal{I}_t$;\n   \\STATE{} $\\hat{\\phi}_{t+1}=\\hat{\\phi}_t-\\frac{1}{n}\\sum_{i_t\\in \\mathcal{I}_t} \\big(\\hat{\\nabla} f_{i_t}(z^t_{i_t})-\\hat{\\nabla} f_{i_t}(z^{t+1}_{i_t})\\big)$;\n   \\ENDFOR\n   \\STATE {\\bfseries Output:} $\\{x,y_{[k]},\\lambda\\}$ chosen at random uniformly from $\\{x_{t},y_{[k]}^{t},\\lambda_t\\}_{t=1}^{T}$.\n\\end{algorithmic}\n\\end{algorithm}\nIn the problem \\eqref{eq:1}, the explicit expression of objective function $f_i(x)$ is not available, and\nonly the function value of $f_i(x)$ is available. To avoid\ncomputing explicit gradient, thus, we use the coordinate smoothing gradient estimator\n\\cite{liu2018zeroth} to estimate gradients: for $i\\in [n]$,\n\\begin{align} \\label{eq:5}\n \\hat{\\nabla} f_i(x) = \\sum_{j=1}^d \\frac{1}{2\\mu_j}\\big(\n f_i(x+\\mu_j e_j)-f_i(x-\\mu_j e_j)\\big)e_j,\n\\end{align}\nwhere $\\mu_j$ is a coordinate-wise smoothing parameter, and $e_j$ is\na standard basis vector with 1 at its $j$-th coordinate, and 0\notherwise.\n\nBased on the above estimated gradients, we propose a zeroth-order\nADMM (ZO-ADMM) method to solve the problem \\eqref{eq:1} by executing the following iterations,\nfor $t \\!=\\! 1,2,\\cdots$\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n& y_j^{t+1} = \\arg\\min_{y_j} \\big\\{ \\mathcal {L}_{\\rho}(x_t,y_{[j-1]}^{t+1},y_j,y_{[j+1:k]}^{t},\\lambda_t) \\\\\n& \\quad \\qquad \\qquad \\qquad + \\frac{1}{2}\\|y_j-y_j^t\\|^2_{H_j} \\big\\}, \\ \\forall j\\in [k] \\\\\n& x_{t+1} = \\arg\\min_x \\hat{\\mathcal {L}}_{\\rho}(x,y_{t+1},\\lambda_t,\\hat{\\nabla} f(x)) \\\\\n& \\lambda_{t+1} = \\lambda_t - \\rho(Ax_{t+1}+By_{t+1}-c),\n\\end{aligned} \\right.\n\\end{equation*}\nwhere the term $\\frac{1}{2}\\|y_j-y_j^t\\|^2_{H_j}$ with $H_j\\succ 0$ to linearize the term $\\|Ax + \\sum_{j=1}^kB_jy_j-c\\|^2$.\nHere, due to using the inexact zeroth-order gradient to update $x$, we define an approximate function over $x_t$ as follows:\n\\begin{align}\n\\small\n & \\hat{\\mathcal {L}}_{\\rho}\\big(x,y_{[k]}^{t+1},\\lambda_t,\\hat{\\nabla} f(x)\\big) \\!=\\!\n f(x_t) \\!+\\! \\hat{\\nabla}f(x)^T(x-x_t) \\nonumber \\\\\n & \\!+\\! \\frac{1}{2\\eta}\\|x\\!-\\!x_t\\|^2_G \\!+\\! \\sum_{j=1}^k \\psi_j(y_j^{t+1}) \\!-\\! \\lambda_t^T(Ax \\!+\\! \\sum_{j=1}^k B_jy_j^{t+1}\\!-\\!c) \\nonumber \\\\\n & \\!+\\! \\frac{\\rho}{2}\\|Ax \\!+\\! \\sum_{j=1}^kB_jy_j^{t+1}\\!-\\!c\\|^2,\n\\end{align}\nwhere $G\\succ0$, $\\hat{\\nabla} f(x)$ is the zeroth-order gradient and $\\eta>0$ is a step size.\nConsidering the matrix $A^TA$ is large, set $G = r I - \\rho \\eta A^TA \\succ I$ with $r > \\rho \\eta\n\\sigma_{\\max}(A^TA) + 1 $ to linearize the term $\\|Ax + \\sum_{j=1}^kB_jy_j^{t+1}-c\\|^2$.\nIn the problem \\eqref{eq:1}, not only the noisy\ngradient of $f_i(x)$ is not available, but also the sample size $n$\nis very large. Thus, we propose fast ZO-SVRG-ADMM and ZO-SAGA-ADMM to solve the problem \\eqref{eq:1},\nbased on the SVRG and SAGA, respectively.\n\nAlgorithm \\ref{alg:1} shows the algorithmic framework of ZO-SVRG-ADMM.\nIn Algorithm \\ref{alg:1}, we use the estimated stochastic gradient\n$\\hat{g}_{t}^{s} = \\hat{\\nabla} f_{\\mathcal{I}_t}(x_{t}^{s})-\\hat{\\nabla} f_{\\mathcal{I}_t}(\\tilde{x}^s)+\\hat{\\nabla} f(\\tilde{x}^s)$\nwith $\\hat{\\nabla} f_{\\mathcal{I}_t}(x^s_t) = \\frac{1}{b}\\sum_{i_t\\in \\mathcal{I}_t} \\hat{\\nabla} f_{i_t}(x^s_t)$.\nWe have $\\mathbb{E}_{\\mathcal{I}_t}[\\hat{g}_{t}^{s}] = \\hat{\\nabla} f(x_{t}^{s}) \\neq \\nabla f(x_{t}^{s})$, \\emph{i.e.}, this stochastic gradient\nis a \\textbf{biased} estimate of the true full gradient.\nAlthough the SVRG has shown a great promise,\nit relies upon the assumption that the stochastic\ngradient is an \\textbf{unbiased }estimate of true full gradient.\nThus, adapting the similar ideas of SVRG to zeroth-order ADMM optimization\nis not a trivial task. To handle this challenge, we choose the appropriate step size\n$\\eta$, penalty parameter $\\rho$ and smoothing parameter $\\mu$ to guarantee the convergence of our algorithms,\nwhich will be discussed in the following convergence analysis.\n\nAlgorithm \\ref{alg:2} shows the algorithmic framework of ZO-SAGA-ADMM.\nIn Algorithm \\ref{alg:2}, we use the estimated stochastic gradient $\\hat{g}_{t} = \\frac{1}{b} \\sum_{i_t\\in \\mathcal{I}_t}\n\\big(\\hat{\\nabla} f_{i_t}(x_{t})-\\hat{\\nabla} f_{i_t}(z^t_{i_t}) \\big) + \\hat{\\phi}_t$\nwith $\\hat{\\phi}_t=\\frac{1}{n}\\sum_{i=1}^n \\hat{\\nabla} f_i(z^t_i)$.\nSimilarly, we have $\\mathbb{E}_{\\mathcal{I}_t}[\\hat{g}_{t}] = \\hat{\\nabla} f(x_t) \\neq \\nabla f(x_t)$.\n\\section{Convergence Analysis}\nIn this section, we will study the convergence properties of\nthe proposed algorithms (ZO-SVRG-ADMM and ZO-SAGA-ADMM).\n\\subsection{Convergence Analysis of ZO-SVRG-ADMM }\nIn this subsection, we analyze convergence properties of the ZO-SVRG-ADMM.\n\nGiven the sequence $\\{(x^{s}_t,y_{[k]}^{s,t},\\lambda^{s}_t)_{t=1}^m\\}_{s=1}^S$ generated from Algorithm \\ref{alg:1},\nwe define a \\emph{Lyapunov} function:\n \\begin{align}\n R^s_t \\! = & \\mathbb{E}\\big[\\mathcal{L}_{\\rho} (x^s_t,y_{[k]}^{s,t},\\lambda^s_t) \\!+\\! (\\frac{3\\sigma^2_{\\max}(G)}{\\sigma^A_{\\min}\\eta^2\\rho} \\!+\\! \\frac{9L^2}{\\sigma^A_{\\min}\\rho})\\|x^s_{t}\\!-\\!x^s_{t-1}\\|^2 \\nonumber \\\\\n & \\quad + \\frac{18 L^2d }{\\sigma^A_{\\min}\\rho b}\\|x^s_{t-1}-\\tilde{x}^s\\|^2 + c_t\\|x^s_{t}-\\tilde{x}^s\\|^2\\big], \\nonumber\n \\end{align}\n where the positive sequence $\\{c_t\\}$ satisfies\n \\begin{equation*}\n  c_t= \\left\\{\n  \\begin{aligned}\n  & \\frac{36 L^2d }{\\sigma^A_{\\min}\\rho b} +\n     \\frac{2Ld}{b} + (1+\\beta)c_{t+1}, \\ 1 \\leq t \\leq m, \\\\\n  & 0, \\ t \\geq m+1.\n  \\end{aligned}\n  \\right.\\end{equation*}\nNext, we definite a useful variable $\\theta^s_{t} \\!=\\! \\mathbb{E} \\big[ \\|x^s_{t+1}-x^s_{t}\\|^2 + \\|x^s_{t}-x^s_{t-1}\\|^2 + \\frac{d}{b}(\\|x^s_{t}-\\tilde{x}^s\\|^2 + \\|x^s_{t-1}-\\tilde{x}^s\\|^2 )\n   + \\sum_{j=1}^k \\|y_j^{s,t}-y_j^{s,t+1}\\|^2\\big]$.\n\\begin{theorem} \\label{th:1}\n Suppose the sequence $\\{(x^{s}_t,y_{[k]}^{s,t},\\lambda^{s}_t)_{t=1}^m\\}_{s=1}^S$ is generated from Algorithm \\ref{alg:1}. Let $m=n^{\\frac{1}{3}}$,\n $b=d^{1-l} n^{\\frac{2}{3}},\\ l \\in\\{ 0,\\frac{1}{2},1\\}$, $\\eta = \\frac{\\alpha\\sigma_{\\min}(G)}{9d^lL} \\ (0 < \\alpha \\leq 1 )$ and\n $\\rho = \\frac{6\\sqrt{71}\\kappa_G d^l L}{\\sigma^A_{\\min}\\alpha}$,\n then we have\n \\begin{align}\n\\min_{s,t} \\mathbb{E}\\big[ \\mbox{dist}(0,\\partial L(x^s_t,y_{[k]}^{s,t},\\lambda^s_t))^2\\big] \\leq O(\\frac{\\tilde{\\nu} d^{2l}}{T}) + O(d^{2+2l}\\mu^2), \\nonumber\n\\end{align}\nwhere $\\tilde{\\nu}=R^1_0 - R^*$, and $R^*$ is a lower bound of function $R^s_t$.\nIt follows that suppose the smoothing parameter $\\mu$ and the whole iteration number $T=mS$ satisfy\n \\begin{align}\n  \\frac{1}{\\mu} = O \\big( \\frac{d^{1+l}}{\\sqrt{\\epsilon}} \\big), \\quad\n   T = O\\big( \\frac{\\tilde{\\nu} d^{2l}}{\\epsilon} \\big), \\nonumber\n \\end{align}\n then $(x^{s^*}_{t^*},y_{[k]}^{s^*,t^*},\\lambda^{s^*}_{t^*})$\n is an $\\epsilon$-approximate stationary point of the problems \\eqref{eq:1}, where $(t^*,s^*) = \\mathop{\\arg\\min}_{t,s}\\theta^s_{t}$.\n\\end{theorem}\n\\begin{remark}\nTheorem \\ref{th:1} shows that given $m=n^{\\frac{1}{3}}$,\n $b=d^{1-l} n^{\\frac{2}{3}},\\ l \\in\\{ 0,\\frac{1}{2},1\\}$, $\\eta = \\frac{\\alpha\\sigma_{\\min}(G)}{9d^lL} \\ (0 < \\alpha \\leq 1 )$,\n $\\rho = \\frac{6\\sqrt{71}\\kappa_G d^l L}{\\sigma^A_{\\min}\\alpha}$ and $\\mu= O(\\frac{1}{d\\sqrt{T}})$,\n the ZO-SVRG-ADMM has convergence rate of $O(\\frac{d^{2l}}{T})$. Specifically, when $1\\leq d < n^{\\frac{1}{3}}$, given $l=0$,\n the ZO-SVRG-ADMM has convergence rate of $O(\\frac{1}{T})$; when $n^{\\frac{1}{3}} \\leq d < n^{\\frac{2}{3}}$, given $l=\\frac{1}{2}$,\n it has convergence rate of $O(\\frac{\\sqrt{d}}{T})$; when $n^{\\frac{2}{3}} \\leq d$, given $l=1$,\n it has convergence rate of $O(\\frac{d}{T})$.\n\\end{remark}\n\\subsection{Convergence Analysis of ZO-SAGA-ADMM }\nIn this subsection, we provide the convergence analysis of\nthe ZO-SAGA-ADMM.\n\nGiven the sequence $\\{x_t,y_{[k]}^{t},\\lambda_t\\}_{t=1}^T$ generated from Algorithm \\ref{alg:2},\nwe define a \\emph{Lyapunov} function\n \\begin{align}\n & \\Omega_t \\! = \\mathbb{E}\\big[ \\mathcal{L}_{\\rho} (x_t,y_{[k]}^{t},\\lambda_t) \\!+\\! (\\frac{3\\sigma^2_{\\max}(G)}{\\sigma^A_{\\min}\\rho\\eta^2}\n  \\!+\\! \\frac{9L^2}{\\sigma^A_{\\min}\\rho}) \\|x_{t}\\!-\\!x_{t-1}\\|^2  \\nonumber \\\\\n &\\quad + \\frac{18 L^2d }{\\sigma^A_{\\min}\\rho b}\\frac{1}{n} \\sum_{i=1}^n\\|x_{t-1}-z^{t-1}_i\\|^2 + c_t\\frac{1}{n} \\sum_{i=1}^n\\|x_{t}-z^t_i\\|^2 \\big]. \\nonumber\n\\end{align}\nHere the positive sequence $\\{c_t\\}$ satisfies\n \\begin{equation*}\n c_t \\!= \\left\\{\n  \\begin{aligned}\n  & \\frac{36L^2d }{\\sigma^A_{\\min}\\rho b} \\!+ \\frac{2Ld}{b} \\!+ (1-\\hat{p})(1+\\beta)c_{t+1}, \\ 0 \\leq t \\leq T-1, \\\\\n  & 0, \\ t \\geq T,\n  \\end{aligned}\n  \\right.    \\end{equation*}\n where $\\hat{p}$ denotes probability of an index $i$ in $\\mathcal{I}_t$.\nNext, we definite a useful variable $\\theta_{t} \\!=\\! \\mathbb{E} \\big[\\|x_{t+1}\\!-\\!x_t\\|^2 \\!+\\! \\|x_t\\!-\\!x_{t-1}\\|^2\n\\!+\\! \\frac{d}{bn}\\sum^n_{i=1} (\\|x_t\\!-\\!z^t_i\\|^2 \\!+\\! \\|x_{t-1}\\!-\\!z^{t-1}_i\\|^2)\\!+\\! \\sum_{j=1}^k \\|y_j^{t}\\!-\\!y_j^{t+1}\\|^2 \\big]$.\n\\begin{theorem} \\label{th:2}\nSuppose the sequence $\\{x_t,y_{[k]}^{t},\\lambda_t\\}_{t=1}^T$ is generated from Algorithm \\ref{alg:2}. Let $b = n^{\\frac{2}{3}}d^{\\frac{1-l}{3}},\\ l \\in\\{ 0,\\frac{1}{2},1\\}$,\n$\\eta = \\frac{\\alpha\\sigma_{\\min}(G)}{33d^lL} \\ (0 < \\alpha \\leq 1)$ and $\\rho = \\frac{6\\sqrt{791}\\kappa_G d^l L}{\\sigma^A_{\\min}\\alpha}$\nthen we have\n\\begin{align}\n\\min_{1\\leq t \\leq T} \\mathbb{E}\\big[ \\mbox{dist}(0,\\partial L(x_t,y_{[k]}^{t},\\lambda_t))^2\\big] \\leq O(\\frac{\\tilde{\\nu}d^{2l}}{T}) \\!+ O(d^{2+2l}\\mu^2), \\nonumber\n\\end{align}\nwhere $\\tilde{\\nu}=\\Omega_0- \\Omega^*$, and $\\Omega^*$ is a lower bound of function $\\Omega_t$.\nIt follows that suppose the parameters $\\mu$ and $T$ satisfy\n\\begin{align}\n \\frac{1}{\\mu} = O \\big( \\frac{d^{1+l} }{\\sqrt{\\epsilon}} \\big), \\quad  T = O\\big(\\frac{\\tilde{\\nu}d^{2l}}{\\epsilon} \\big), \\nonumber\n\\end{align}\nthen $(x_{t^*},y_{[k]}^{t^*},\\lambda_{t^*})$ is an $\\epsilon$-approximate stationary point of the problems \\eqref{eq:1},\nwhere $t^* = \\mathop{\\arg\\min}_{ 1\\leq t\\leq T}\\theta_{t}$.\n\\end{theorem}\n\\begin{remark}\nTheorem \\ref{th:2} shows that $b = n^{\\frac{2}{3}}d^{\\frac{1-l}{3}},\\ l \\in\\{ 0,\\frac{1}{2},1\\}$,\n$\\eta = \\frac{\\alpha\\sigma_{\\min}(G)}{33d^lL} \\ (0 < \\alpha \\leq 1)$, $\\rho = \\frac{6\\sqrt{791}\\kappa_G d^l L}{\\sigma^A_{\\min}\\alpha}$\nand $\\mu= O(\\frac{1}{d\\sqrt{T}})$, the ZO-SAGA-ADMM has the $O(\\frac{d^{2l}}{T})$ of convergence rate. Specifically, when $1\\leq d < n$, given $l=0$,\n the ZO-SAGA-ADMM has convergence rate of $O(\\frac{1}{T})$; when $n \\leq d < n^2$, given $l=\\frac{1}{2}$,\n it has convergence rate of $O(\\frac{d}{T})$; when $n^2 \\leq d$, given $l=1$,\n it has convergence rate of $O(\\frac{d^2}{T})$.\n\\end{remark}\n\\begin{figure*}[!t]\n\\centering\n\\subfigure[20news]{\\includegraphics[width=0.235\\textwidth]{20news_obj}}\n\\subfigure[a9a]{\\includegraphics[width=0.235\\textwidth]{a9a_obj}}\n\\subfigure[w8a]{\\includegraphics[width=0.235\\textwidth]{w8a_obj}}\n\\subfigure[covtype.binary]{\\includegraphics[width=0.235\\textwidth]{covtype_obj}}\n\\caption{Objective value gaps \\emph{versus} CPU time on benchmark datasets.}\n\\label{fig:1}\n\\end{figure*}\n\\section{Experiments}\nIn this section, we compare our algorithms\n(ZO-SVRG-ADMM, ZO-SAGA-ADMM) with the ZO-ProxSVRG, ZO-ProxSAGA \\cite{Huang2019faster}, the deterministic zeroth-order ADMM (ZO-ADMM),\nand zeroth-order stochastic ADMM (ZO-SGD-ADMM) without variance reduction\non two applications: 1) robust black-box binary classification, and 2) structured adversarial\nattacks on black-box DNNs.\n\\begin{table}[!h]\n  \\centering\n  \\begin{tabular}{c|c|c|c}\n  \\hline\n  datasets & \\#samples & \\#features & \\#classes \\\\ \\hline\n  \\emph{20news} & 16,242 &  100 & 2 \\\\\n  \\emph{a9a}   & 32,561 & 123 & 2 \\\\\n  \\emph{w8a} & 64,700 & 300 & 2 \\\\\n  \\emph{covtype.binary} & 581,012 & 54 & 2\\\\\n  \\hline\n  \\end{tabular}\n  \\caption{Real Datasets for Black-Box Binary Classification }\\label{tab:2}\n\\end{table}\n\n\\begin{figure*}[!t]\n  \\centering\n \n  \\includegraphics[width=0.75\\textwidth]{Adversarial_attacks}\\\\\n  \\caption{Group-sparsity perturbations are learned from MNIST and CIFAR-10 datasets. Blue and red labels denote the initial label,\n  and the label after attack, respectively. }\\label{fig:3}\n\\end{figure*}\n\\subsection{ Robust Black-Box Binary Classification }\nIn this subsection, we focus on a robust black-box binary classification task\nwith graph-guided fused lasso.\nGiven a set of training samples $(a_i,l_i)_{i=1}^n$,\nwhere $a_i\\in \\mathbb{R}^d$ and $l_i \\in \\{-1,+1\\}$,\nwe find the optimal parameter $x\\in \\mathbb{R}^d$ by solving the problem:\n\\begin{align} \\label{eq:8}\n \\min_{x\\in \\mathbb{R}^d} \\frac{1}{n}\\sum_{i=1}^n f_i(x) + \\tau_1\\|x\\|_1 + \\tau_2\\|\\hat{G}x\\|_1,\n\\end{align}\nwhere $f_i(x)$ is the black-box loss function,\nthat only returns the function value given an input.\nHere, we specify the loss function\n$f_i(x)= \\frac{\\sigma^2}{2}\\big(1-\\exp(-\\frac{(l_i-a_i^Tx)^2}{\\sigma^2})\\big)$,\nwhich is the \\emph{nonconvex} robust correntropy\ninduced loss \\cite{he2011maximum}.\nMatrix $\\hat{G}$ decodes the sparsity pattern of graph obtained by learning sparse Gaussian graphical model \\cite{huang2015joint}.\nIn the experiment, we give mini-batch size $b=20$,\nsmoothing parameter $\\mu=\\frac{1}{d\\sqrt{t}}$ and penalty parameters $\\tau_1=\\tau_2=10^{-5}$.\n\nIn the experiment, we use\nsome public real datasets\\footnote{\\emph{20news} is from \\url{https:\/\/cs.nyu.edu\/~roweis\/data.html};\nothers are from \\url{www.csie.ntu.edu.tw\/~cjlin\/libsvmtools\/datasets\/}.},\nwhich are summarized in Table \\ref{tab:2}. For each dataset, we use half of the samples as training data\nand the rest as testing data. Figure \\ref{fig:1} shows that the objective values\nof our algorithms faster decrease than the\nother algorithms, as the CPU time increases. In particular, our algorithms show better performances than\nthe zeroth-order proximal algorithms. It is relatively difficult that these zeroth-order proximal methods deal with\nthe nonsmooth penalties in the problem \\eqref{eq:8}.\nThus, we have to use some iterative methods\nto solve the proximal operator in these proximal methods.\n\\subsection{Structured Attacks on Black-Box DNNs }\nIn this subsection, we use our algorithms to generate adversarial examples to attack the pre-trained DNN models,\nwhose parameters are hidden from us and only its outputs are accessible.\nMoreover, we consider an interesting problem: ``What possible\nstructures could adversarial perturbations have to fool black-box DNNs ?\"\nThus, we use the zeroth-order algorithms to find an universal structured adversarial perturbation $x \\in \\mathbb{R}^d$\nthat could fool the samples $\\{a_i\\in \\mathbb{R}^d , \\ l_i \\in \\mathbb{N} \\}_{i=1}^n$,\nwhich can be regarded as the following problem:\n\\begin{align} \\label{eq:9}\n\\min_{x \\in \\mathbb{R}^d} &\\frac{1}{n}\\sum\\limits_{i=1}^n \\max\\big\\{F_{l_i}(a_i+x) - \\max\\limits_{j\\neq l_i} F_j(a_{i}+x), 0 \\big\\} \\nonumber \\\\\n &  + \\tau_1 \\sum_{p=1}^P\\sum_{q=1}^Q \\|x_{\\mathcal{G}_{p,q}}\\|_2 + \\tau_2 \\|x \\|_2^2 + \\tau_3 h(x),\n\\end{align}\nwhere $F(a)$ represents the final layer output before softmax of neural network, and $h(x)$ ensures the validness of created adversarial examples.\nSpecifically, $h(x) = 0$ if $a_i+x \\in [0,1]^d$ for all $i\\in [n]$ and $\\|x\\|_\\infty \\leq \\epsilon$, otherwise $h(x) = \\infty$.\nFollowing \\cite{xu2018structured}, we use the overlapping lasso to obtain structured perturbations.\nHere, the overlapping groups $\\{\\mathcal{G}_{p,q}\\},\\ p=1,\\cdots,P, \\ q = 1,\\cdots,Q$\ngenerate from dividing an image into\nsub-groups of pixels.\n\nIn the experiment, we use the pre-trained DNN models on MNIST and CIFAR-10 as the target black-box models, which can attain $99.4\\%$ and $80.8\\%$ test accuracy, respectively.\nFor MNIST, we select 20 samples from a target class and set batch size $b=4$; For CIFAR-10, we select 30 samples and set $b=5$.\nIn the experiment, we set $\\mu=\\frac{1}{d\\sqrt{t}}$, where $d=28 \\times 28$ and $d=3\\times 32\\times 32$ for MNIST and CIFAR-10, respectively.\nAt the same time, we set the parameters $\\epsilon=0.4$, $\\tau_1 = 1$, $\\tau_2=2$ and $\\tau_3=1$.\nFor both datasets, the kernel size for overlapping group lasso is set to $3 \\times 3$ and the stride is one.\n\nFigure \\ref{fig:2} shows that attack losses (\\emph{i.e.} the first term of the problem \\eqref{eq:9}) of\nour methods faster decrease than the other methods, as the number of iteration increases.\nFigure \\ref{fig:3} shows that our algorithms can learn some structure perturbations,\nand can successfully attack the corresponding DNNs.\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[\\emph{MNIST}]{\\includegraphics[width=0.236\\textwidth]{Mnist_attack_loss}}\n\\subfigure[\\emph{CIFAR-10}]{\\includegraphics[width=0.236\\textwidth]{CIFAR_attack_loss}}\n\\caption{Attack loss on adversarial attacks black-box DNNs. }\n\\label{fig:2}\n\\end{figure}\n\\section{Conclusions}\nIn the paper, we proposed fast ZO-SVRG-ADMM and ZO-SAGA-ADMM methods based on\nthe coordinate smoothing gradient estimator, which only uses the objective function\nvalues to optimize. Moreover, we prove that the proposed methods have a convergence rate of $O(\\frac{1}{T})$.\nIn particular, our methods not only reach the existing best convergence rate $O(\\frac{1}{T})$ for the nonconvex optimization,\nbut also are able to effectively solve many machine learning problems with the complex nonsmooth regularizations.\n\\section*{Acknowledgments}\nF.H., S.G., H.H. were partially supported by\nU.S. NSF IIS 1836945, IIS 1836938, DBI 1836866, IIS 1845666, IIS 1852606, IIS 1838627, IIS 1837956.\nS.C. was partially supported by the NSFC under Grant No. 61806093 and No. 61682281, and the Key Program of NSFC under Grant No. 61732006.\n\n\n\\bibliographystyle{named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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Author names \nshould be listed starting from left to right and then moving down to the \nnext line. This is the author sequence that will be used in future citations \nand by indexing services. Names should not be listed in columns nor group by \naffiliation. Please keep your affiliations as succinct as possible (for \nexample, do not differentiate among departments of the same organization).\n\n\\subsection{Identify the Headings}\nHeadings, or heads, are organizational devices that guide the reader through \nyour paper. There are two types: component heads and text heads.\n\nComponent heads identify the different components of your paper and are not \ntopically subordinate to each other. Examples include Acknowledgments and \nReferences and, for these, the correct style to use is ``Heading 5''. Use \n``figure caption'' for your Figure captions, and ``table head'' for your \ntable title. Run-in heads, such as ``Abstract'', will require you to apply a \nstyle (in this case, italic) in addition to the style provided by the drop \ndown menu to differentiate the head from the text.\n\nText heads organize the topics on a relational, hierarchical basis. For \nexample, the paper title is the primary text head because all subsequent \nmaterial relates and elaborates on this one topic. If there are two or more \nsub-topics, the next level head (uppercase Roman numerals) should be used \nand, conversely, if there are not at least two sub-topics, then no subheads \nshould be introduced.\n\n\\subsection{Figures and Tables}\n\\paragraph{Positioning Figures and Tables} Place figures and tables at the top and \nbottom of columns. Avoid placing them in the middle of columns. Large \nfigures and tables may span across both columns. Figure captions should be \nbelow the figures; table heads should appear above the tables. Insert \nfigures and tables after they are cited in the text. Use the abbreviation \n``Fig.~\\ref{fig}'', even at the beginning of a sentence.\n\n\\begin{table}[htbp]\n\\caption{Table Type Styles}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\textbf{Table}&\\multicolumn{3}{|c|}{\\textbf{Table Column Head}} \\\\\n\\cline{2-4} \n\\textbf{Head} & \\textbf{\\textit{Table column subhead}}& \\textbf{\\textit{Subhead}}& \\textbf{\\textit{Subhead}} \\\\\n\\hline\ncopy& More table copy$^{\\mathrm{a}}$& &  \\\\\n\\hline\n\\multicolumn{4}{l}{$^{\\mathrm{a}}$Sample of a Table footnote.}\n\\end{tabular}\n\\label{tab1}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics{fig1.png}}\n\\caption{Example of a figure caption.}\n\\label{fig}\n\\end{figure}\n\nFigure Labels: Use 8 point Times New Roman for Figure labels. Use words \nrather than symbols or abbreviations when writing Figure axis labels to \navoid confusing the reader. As an example, write the quantity \n``Magnetization'', or ``Magnetization, M'', not just ``M''. If including \nunits in the label, present them within parentheses. Do not label axes only \nwith units. In the example, write ``Magnetization (A\/m)'' or ``Magnetization \n\\{A[m(1)]\\}'', not just ``A\/m''. Do not label axes with a ratio of \nquantities and units. For example, write ``Temperature (K)'', not \n``Temperature\/K''.\n\n\\section*{Acknowledgment}\n\nThe preferred spelling of the word ``acknowledgment'' in America is without \nan ``e'' after the ``g''. Avoid the stilted expression ``one of us (R. B. \nG.) thanks $\\ldots$''. Instead, try ``R. B. G. thanks$\\ldots$''. Put sponsor \nacknowledgments in the unnumbered footnote on the first page.\n\n\\section*{References}\n\nPlease number citations consecutively within brackets \\cite{b1}. The \nsentence punctuation follows the bracket \\cite{b2}. Refer simply to the reference \nnumber, as in \\cite{b3}---do not use ``Ref. \\cite{b3}'' or ``reference \\cite{b3}'' except at \nthe beginning of a sentence: ``Reference \\cite{b3} was the first $\\ldots$''\n\nNumber footnotes separately in superscripts. Place the actual footnote at \nthe bottom of the column in which it was cited. Do not put footnotes in the \nabstract or reference list. Use letters for table footnotes.\n\nUnless there are six authors or more give all authors' names; do not use \n``et al.''. Papers that have not been published, even if they have been \nsubmitted for publication, should be cited as ``unpublished'' \\cite{b4}. Papers \nthat have been accepted for publication should be cited as ``in press'' \\cite{b5}. \nCapitalize only the first word in a paper title, except for proper nouns and \nelement symbols.\n\nFor papers published in translation journals, please give the English \ncitation first, followed by the original foreign-language citation \\cite{b6}.\n\n\n\\section{Introduction}\n\nCritically ill patients in the intensive care unit (ICU) commonly experience pain because of a multitude of factors including surgery, trauma, burns, and cancer, as well as enduring procedural pain \\cite{Barr2013}. Regardless of source, poorly managed pain adversely affects multiple organ systems and has a direct impact on outcomes, as the negative physiologic and psychologic consequences of pain are significant, long lasting, and can  preclude patients from participating in their care.\n\nIn the ICU, opioids are the primary systemic medications for managing pain. The optimal choice of opioid and the dosing regimen used for an individual patient depend on many factors, including the intensity of pain. To assess the latter as well as to measure  response to treatment, multiple pain intensity scales have been developed. These are unidimensional scales that measure intensity from 0 (no pain) to 10 (worst possible pain).\nUnfortunately, pain assessment scales fail in multiple patient populations (e.g. non-verbal children) and clinical situations (e.g. during paralysis). Hence, in recent years, an increasing body of work is addressing the problem of pain measurement using machine leaning with autonomic  \\cite{dlmEMBC2018}, brain activity \\cite{dlm_ICPR_2018} and facial expression \\cite{LopezMartinez2017c} data, and by stratifying patients based on their pain responses  to allow for personalized assessments \\cite{dlmNIPS2017}. However, machine learning for automatic pain management interventions remains an unexplored area.\n\n\nIn this work, we propose a novel approach for the automatic dosing of analgesics based on deep reinforcement learning. Specifically, we focus on morphine, one of the most commonly prescribed opioids in the ICU and a first line for the treatment of pain. Both under-treatment and over-treatment with morphine lead to severe adverse effects, such as those related to unrelieved pain and respiratory depression respectively. Our model aims to learn clinically interpretable optimal treatment policies and recommend individualized morphine interventions adapted to each patient's evolving state, to minimize pain while avoiding adverse effects.\nTo our knowledge, our work is the first to apply reinforcement learning for the recommendation of pain management regimes and the automatic dosing of analgesics.\n\n\\section{Background and related work}\n\nReinforcement learning (RL) algorithms model time-varying state and action spaces, and learn actionable policies $\\pi$ (mappings from states to actions) from training examples that do not represent optimal behaviour. At each time step $t$, the RL agent observes the current environment state $s_t$ and takes an action $a_t \\in \\mathcal{A}$ from a discrete set of actions $\\mathcal{A}$, subsequently receiving a reward $r_t$ and transitioning to a new state $s_{t+1}$. The action $a_t$ is chosen following policy $\\pi$ so that the return or expected sum of future rewards $R_t = r_t + \\gamma r_{t+1} + \\gamma^2 t_{t+2}+...$ when taking that action and following policy $\\pi$ thereafter is maximized. The future rewards are discounted by a factor $\\gamma$ that represents the trade off between  immediate and later rewards.\n\nCombining RL with deep neural networks has achieved tremendous success in complex problems such as the Atari, Go, and chess games \\cite{Mnih2015, Silver2018}. In medicine, deep reinforcement learning has been applied, for example, to unfractioned heparin dosing in the ICU \\cite{Nemati2016}, intravenous fluid and vasopressor dosing in sepsis patients \\cite{Raghu2017,Raghu2017a}, and chemotherapy dosing in cancer patients enrolled in clinical trials \\cite{Yauney2018}.\n\n\n\n\n\\section{Methods}\nMorphine  analgesia is characterized by tremendous interindividual variation and is influenced by multiple factors \\cite{Fillingim2005}. Hence, in the ICU, physicians continuously monitor patients to personalize morphine pain management. Our goal is to deduce optimal treatment policies, similar to those of physicians, that account for pain level as well as physiological state as captured by vital signs. \nWe chose RL over supervised learning because no ground truth for treatment strategies exists and RL is able to infer optimal strategies from suboptimal training examples.\n\n\\subsection{Data and preprocessing}\n\nThis work was  conducted on the publicly available Multiparameter Intelligent Monitoring in Intensive Care III (MIMIC-3) database \\cite{Johnson2016}, which contains hospital admissions from approximately 38600 individuals aged $\\geq 15$ years. From MIMIC-3, we extracted a cohort of ICU patients fulfilling 2 criteria: (1) at least one pain intensity score  had been collected, and (2) at least one dose of intravenous morphine  had been administered.\nA total of 6843 admissions fulfilled these criteria. We chose to focus on morphine since this was the most commonly prescribed opioid in the MIMIC-3 database, followed by hydromorphone and oxycodone.\n\n\\subsection{Feature extraction}\n\nFor each patient, we extracted pain scores, analgesic administration events and vital signs. Data were aggregated into non-overlapping one-hour windows. When several data points were available within a window, we replaced them by either the mean (pain scores, vital signs) or the sum (analgesic administrations).  To account for missing data, we utilized sample-and-hold interpolation as in \\cite{Nemati2016}. In total, we extracted 20 hourly features.\n\n\n\\subsubsection{Pain scores}  MIMIC-III  contains scores in both numerical and text format. The latter were converted to a numeric score by using the scores referenced in the text (e.g. \\textit{\"3-Mild to Mod\"} becomes 3). All scores were reported in a 0 (no pain) to 10 (maximum pain) scale.\n\n\n\n\\subsubsection{Morphine interventions}\nMorphine sulfate can be administered intravenously either as a bolus  or as a continuous infusion with a steady rate for an extended period of time. In this work, we considered intravenous boluses only, which represented a majority of administrations for the studied admissions (59.4\\%).\nThese typically have an onset of 5-10 min and a half life of 3-4 hr.  \\cite{Narayanan2016a}. The dose, in addition to time of administration, was recorded.\n\n\n\\subsubsection{Non-morphine analgesic interventions} \nIn order to  lower opioid consumption and to improve the quality of analgesia, morphine is commmonly administered with adjuvant analgesics \\cite{Narayanan2016a}. This includes nonopiod coanalgesics such as nonsteroidal anti-inflammatory agents. Therefore, we also extracted administrations of 16 analgesics in addition to morphine.\n\n\n\n\n\\subsubsection{Physiological signals}\nMorphine induces respiratory depression \\cite{Pattinson2008} and impacts the cardiovascular system \\cite{Chen2015} with potentially fatal consequences.\nHence, we extracted two relevant physiological signals:  respiration rate (RR) and  heart rate (HR).\n\n\n\\subsection{Continuous State-space and Discretized Action-space }\n\\label{sec:spaces}\nThe action space was discretized to 14 actions: $0$, $(0,1]$, $(1,2]$, ..., $(9,10]$, $(10,15]$, $(15,20]$ and $(20,\\infty)$ mg of morphine intravenously. The state space, on the other hand, was continuous and directly  captured the patient's self-reported pain intensity and physiological state, as well as non-morphine analgesic interventions.\n\n\n\\subsection{Reward function}\n\nAt each time stamp $t$, the RL agent interacts with the environment and observes the state $s_t$ to  choose whether to deliver a morphine IV bolus (actions 1-13) or to withhold morphine (action 0). The action $a_t$ is chosen to  maximize the return, that is, the expected discounted future reward, defined as $R_t = \\sum_{t'=t}^{T} \\gamma^{t'-t} r_{t'}$, where $T$ is the terminal timestep.\n\n\nWe defined a reward function $r_t(hr,rr,p)$ that incentivizes \npain reduction while seeking to maintain heart rate and respiration rate within adequate physiological ranges. For healthy adults at rest, heart rates between 60 and 100 beats per minute and respiration rates between 12 and 20 breaths per minute are considered acceptable. \n\\begin{equation}\nr_t(hr,rr,p) = C_\\text{P} \\cdot r^{\\text{P}}_t (p_t)\n    + C_\\text{HR} \\cdot  r^{\\text{HR}}_t (hr_t)\n    + C_\\text{RR} \\cdot   r^{\\text{RR}}_t (rr_t)\n\\end{equation}\nwhere $C_\\text{P} = C_\\text{HR} =  C_\\text{RR} = \\frac{1}{3}$ and \n{\\small\n\\begin{flalign}\n    & r^{\\text{HR}}_t (hr_t) =     \\frac{2}{1 + e^{-({hr}_t- 60)} } \n                        -  \\frac{2}{1 + e^{-({hr}_t-100)} } \n                        -  1  &\\\\\n    & r^{\\text{RR}}_t (rr_t) =    \\frac{2}{1 + e^{-({rr}_t-12)} } \n                       -  \\frac{2}{1 + e^{-({rr}_t-20)} } \n                       -  1    &\\\\ \n    & r^{\\text{P}}_t (p_t) = 1 - \\frac{2 p_t}{10}  & \n\\end{flalign}\n}\nThis function assigns the maximum reward when pain is absent and both heart rate and respiration rate are within the acceptable range, while diminishing as they move away from the ideal range.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{figures\/Example_v3.png}\n\t\\caption{Example of a MIMIC-III patient's time series of physiological data and pain scores, together with the morphine administrations (discretized to 14  actions as described in Sec.\\ref{sec:spaces}).}\n\t\\label{fig:Example}\n\\end{figure}\n\n\\subsection{Reinforcement learning model architecture}\n\n\\begin{comment}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{figures\/Model.png}\n\t\\caption{DQN model used in this research.}\n\t\\label{fig:Model}\n\\end{figure}\n\\end{comment}\n\n\nTo learn the optimal treatment policies, we use Q-learning \\cite{Watkins92}, which works by learning an action-value $Q(s_t,a_t)$ function that provides the expected return $R_t$ of taking a given action at each state, that is, \n$\n    Q(s_t,a_t) = \\max_\\pi \\mathbb{E}[R_t]\n$\nwhere the maximum is taken over all possible policies $\\pi$. In Q-learning, the optimal action-value function is estimated using the Bellman equation: \n{ \\small\n\\begin{equation}\nQ(s_t, a_t) = \\mathbb{E}_{s_{t+1} \\sim T(s_{t+1}|s_t,a_t)}  [r + \\gamma \\max_{a_{t+1}} Q(s_{t+1},a_{t+1}) | s_t, a_t ]\n\\end{equation}\n}\nwhere $T(s_{t+1}|s_t,a_t)$ is the state transition distribution. Deep Q-learning uses neural networks to approximate $Q(s_t,a_t)$ \\cite{Mnih2015} by sampling tuples $<s_t,a_t,r_t,s_{t+1}>$ and \nminimizing the squared error loss between the output and  target $Q$ values, where $Q_{\\mbox{target}} = r_t+ \\gamma \\max_{a_{t+1}} Q (s_{t+1}, a_{t+1}; \\theta)$ and $\\theta$  are the weights used to parametrize the target network, using stochastic gradient descent.\n\n\nIn this work, following \\cite{Raghu2017,Raghu2017a}, we combined two common extensions of the deep Q network (DQN) model: double-deep Q networks (DDQN) \\cite{Hasselt2016} and dueling Q networks (dueling DQN) \\cite{Wang2015}. These address several shortcomings of simple DQNs, such as their tendency to overestimate Q values.\n\\begin{comment}\nDouble DQNs overcome the tendency of DQNs to overestimate Q values by modifying the Bellman update to:\n\\begin{equation}\n    Q(s_t, a_t) = r_t + \\gamma \\max_a Q'(s_{t+1}, \\arg\\max_a Q(s_{t+1},a) )\n\\end{equation}\nDueling DQNs, on the other hand, into separate \\textit{value} and \\textit{advantage} streams, where the \\textit{value} represents the quality of the current state, and the \\textit{advantage} represents the quality of the chosen action, such that:\n\\begin{equation}\nQ(s,a) = V(s) + A(s,a)\n\\end{equation}\n\\end{comment}\n\nOur final network architecture was therefore a Dueling Double-Deep Q Network (Dueling DDQN),\nIt contained two hidden layers with 64 nodes each, with Leaky-ReLu activation functions and equally sized advantage and value streams. As in \\cite{Raghu2017, Raghu2017a}, we used prioritized experience replay to sample transitions from the training set with probability proportional to the previous error.  The network was implemented in TensorFlow.\n\n\n\n\\section{Results}\n\nWe separated  6843 hospital admissions into 70\\%\/20\\%\/10\\% training, validation and test sets respectively. The action space was discretized to 14 actions as described in Sec.\\ref{sec:spaces}, based on visual inspection of the histogram of morphine doses in MIMIC-III.\nThe dueling DDQN model was then trained using Adam optimization \\cite{Kingma2015} with batch size 32. After training, we obtained the optimal policy $\\pi$ for a given patient state $s$, such that $\\pi(s)= \\arg \\max _a Q(s,a)$. Unfortunately, quantitative evaluation of off-policy models is challenging because it is difficult to assess the impact of deploying the learned policy $\\pi$ in patient outcomes. Hence, we evaluated $\\pi$ by qualitatively examining the choices of treatments proposed by the model, and comparing these with those made by physicians and our understanding of accepted clinical guidelines.\n\nFig.\\ref{fig:histogram1} depicts the comparison of actions taken by the physicians with those recommended by the model, for every hour in the test set. Action 0 refers to no morphine administered, whereas actions 1-13 represent increasing morphine doses. Physicians do not often prescribe morphine to patients (only 5.80\\% hourly timestamps contained morphine administration), but when they do, in 99\\% of instances the model also recommended morphine administration. The agreement between physician actions and model recommendations in these instances is shown in Fig.\\ref{fig:2DHistogram}. The actions recommended for the model for the 94.2\\% instances in which physicians chose to withhold morphine are shown in Fig.\\ref{fig:histogram1}.\n\nWhile physicians administer morphine sparsely, when they do, a second morphine dose is administered in the next hour in more than 40\\% of occasions, as shown in Fig.\\ref{fig:1DHistograms}. The model, on the other hand, recommended more continuous administration of morphine. This result is consistent with previous studies comparing intermittent bolus versus continuous infusion dosing of morphine, which have shown that continuous dosing provides similar or even better pain relief with no increase in acute adverse effects \\cite{Yu2014,Rutter1980}.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/combined_v3_tight.png}\n\t\\caption{\\textit{(Left)} Normalized histogram of physician and dueling DDQN actions, where all actions that involved administration of morphine (actions 1-13) were grouped together. \\textit{(Right)} Histogram of dueling DDQN recommended actions for those instances in which physicians chose to withhold morphine (action 0).}\n\t\\label{fig:histogram1}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{figures\/2DHistogram_v3.png}\n\t\\caption{Histogram of physician against dueling DDQN actions  for those instances in which both physician and dueling DDQN recommended  administration of morphine (actions 1-13).}\n\t\\label{fig:2DHistogram}\n\\end{figure}\n\n\n\\section{Conclusion}\nThis work builds on recent research using  reinforcement learning to automatically find optimal treatment policies to improve patient outcome from training examples that do not represent optimal behavior  \\cite{Nemati2016,Raghu2017,Raghu2017a}. Here, we focus on the problem of dosing opioid analgesics in the ICU. While machine learning has been applied extensively to the problem of pain recognition, to our knowledge this is the first application of reinforcement learning\nfor deducing optimal intervention regimens to manage pain. We aimed to demonstrate how deep reinforcement learning could be used to suggest clinically interpretable treatment strategies for pain using real-time patient data that are readily accessible.  \nThis work has many limitations and should be taken as an illustrative example. Morphine analgesia in the ICU is a complex decision problem influenced by multiple factors. However, our model simplified these clinical factors by constraining  the amount of variables considered \nand defining a reward function that only aims to reduce pain while keeping heart and respiration rates within a pre-defined range. Therefore, possible directions for future work include improving the state space by adding more data modalities, extending the action space to include more dosing regimes (e.g. co-analgesics), and increasing the sophistication of the reward function to reflect the broader range of therapeutic goals. Another limitation is that we used data from a single center, which raises concerns about the generalizability of our findings. Hence, future studies should include multiple centers.\nDespite the limitations of this work, our learned policy is consistent with research that shows that continuous dosing is recommended over intermittent dosing. \n\nFinally, deployment of this approach to opioid dosing in clinical settings would first require extensive prospective evaluation of the learned policies. The goal would not be to replace physicians'  clinical judgments about treatment, but to aid clinical decision making with insights about optimal decisions and automatically guide therapy.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/fourfigs_v3_tight.png}\n\t\\caption{Histogram of morphine administration doses \\textit{(top)} and time between administrations \\textit{(bottom)} for both physicians \\textit{(left)} and the dueling DDQN algorithm \\textit{(right)}.}\n\t\\label{fig:1DHistograms}\n\\end{figure}\n\n{\\small\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nAfter a recent discovery of  topological insulators (TIs) \\cite{HasanRMP2010,QiRMP2011}, the quest for numerous fascinating effects in response to external perturbations in TIs has started. One of them is the magnetoelectric (ME) effect, which is a long standing issue in multiferroics \\cite{Tokura2007} and has been recently introduced in TIs \\cite{HasanRMP2010,QiRMP2011,Qi2008,Essin2009,Vazifeh2010,Nomura2011,Tse2010}. The essence of the ME effect is that the external electric or magnetic fields induce magnetization or polarization in a TI, respectively. From the effective topological field theory predictions \\cite{Qi2008}, the ME effect in three-dimensional (3D) TIs can be described by introducing a new term $U_{\\theta}$ in the energy density for the axion electrodynamics \\cite{HasanRMP2010,QiRMP2011,Qi2008,Essin2009,Chen2011,Nomura2011,Tse2010,Wilczek1987, Vazifeh2010,Coh2011,Sitte2012}:\n\\begin{equation}\n\\label{Eq:Enden}\nU_{\\theta}=-\\frac{e^2}{4\\pi^2\\hbar c}\\theta\\mathbf{E}\\cdot\\mathbf{B},\n\\end{equation}\nwhere $\\mathbf{E}$ and $\\mathbf{B}$ are the electric and magnetic fields respectively. The $\\theta$ term, $U_{\\theta}$ \\cite{Wilczek1987}, characterizes the 3D TIs. In Eq.~(\\ref{Eq:Enden}), parameter $\\theta$ takes values $\\pm\\pi$  for TIs, whereas it is $0$ for ordinary insulators.\nThe magnetization $\\mathbf{M}$ and electric polarization $\\mathbf{P}$ are obtained from the energy density as follows:\n\\begin{eqnarray}\n\\mathbf{M}&=&-\\frac{\\partial U_{\\theta}}{\\partial \\mathbf{B}}=\\frac{e^2 }{4\\pi^2\\hbar c}\\theta \\mathbf{E},\\\\\n\\qquad \\mathbf{P}&=&-\\frac{\\partial U_{\\theta}}{\\partial\\mathbf{E}}=\\frac{e^2}{4\\pi^2\\hbar c}\\theta\\mathbf{B}, \n\\end{eqnarray}\nand clearly show the cross-correlated responses \\cite{Tokura2007,Qi2008,Essin2009,Essin2010,Malashevich2010,Malashevich2011}.\n\nIn a strong magnetic field, which breaks time reversal symmetry, however, a controversy still remains in the literature regarding the quantization of ME responses. According to Ref.~\\onlinecite{Tse2010}, which considers only TI surface states, the response should be quantized: $\\theta\/\\pi=2N+1$ ($N$ being an integer) when the electron densities on the top and bottom surfaces are balanced, i.e., no gate voltage is applied. On the other hand, Ref.~\\onlinecite{Coh2011}, considering only bulk states in a magnetic field, concludes that $\\theta$ is not quantized but rather arbitrary. \n\nIn this paper, to resolve this controversy, we study the effect of an ultrahigh magnetic field on TIs. To fully treat both bulk and surfaces, we consider the lattice Hamiltonian \\cite{Zhang2009,Liu2010} for a 3D TI in a slab geometry, sandwiched between two thin ferromagnets with magnetizations pointing in the opposite directions along $z$ axis, see Fig.~\\ref{fig:EM}. The role of ferromagnetic films can also be played by an appropriate doping of the TI with magnetic impurities. \n\\begin{figure}\n\\includegraphics[width=0.99\\columnwidth]{me_TI}\n \\caption{(Color online) The ME effect in a 3D TI of slab geometry sandwiched between two ferromagnets (FMs) with opposite magnetizations.}\n \\label{fig:EM}\n\\end{figure}\nWe first reproduce the $\\theta$ term (\\ref{Eq:Enden}) with $\\theta=\\pm\\pi$, in the balanced case, from the total energy of electrons. This means that the axion electrodynamics remains unchanged as long as the bulk gap is not destroyed, even though time-reversal symmetry, which plays essential role in TIs, is broken by the magnetic field. However, as the TI thickness is decreased, we observe that the ME effect starts to vanish. Based on this result, we estimate the critical thickness of TI film capable of displaying the robust ME effect, which can be useful for future experimental studies, especially in the view of recent experimental progress on TI thin films \\cite{Checkelsky2012,Chang2013}. Moreover, we also study the case of both strong electric and magnetic fields. We show that the response to a small change in the electric field can be characterized in this regime by different $\\theta$, which are \\textit{still quantized} as long as the Fermi level resides within the bulk gap. We show that the quantization rule of $\\theta$ is determined by Landau level structures on the top and bottom surfaces, while the bulk states do not play an important role in thick TI films.\n\n\\section{Model}\n\nTo study the ME effect in TIs, we employ the Wilson-Dirac tight-binding Hamiltonian \nfor 3D lattice TI model \\cite{Zhang2009,Liu2010}, which is a simplified version of the effective four-band Hamiltonian for newly discovered 3D TIs such as $\\mathrm{Bi}_2\\mathrm{Se}_3,\\mathrm{Bi}_2\\mathrm{Te}_3$, and $\\mathrm{Sb}_2\\mathrm{Te}_3$ \\cite{Xia2009,Chen2009,Hsieh2009}. We introduce external magnetic field $\\mathbf{B}$, applied in $z$ direction, through the Peierls substitution and set its magnitude $B$ so that the magnetic flux going through a single plaquette is a rational fraction of the magnetic flux quantum. In other words, $B=(p\/q)\\Phi_0\/a^2$, where $p$ and $q$ are natural numbers with $p<q$, $\\Phi_0=hc\/e$ is the magnetic flux quantum, and $a$ is the lattice constant. For convenience, we set $e=c=\\hbar=a=1$ in the following so the magnetic field becomes $B=2\\pi p\/q$. Using Landau gauge $\\mathbf{A}=(0,Bx,0)$, the Hamiltonian is defined as \n\\begin{equation}\t\t\n\\label{eq:Wlsn}\t\nH_0=-\\frac{1}{2}\\sum_{\\mu=x,y,z}(T_{\\mu}^{}+T_{\\mu}^{\\dag})+(m+3r)\\sum_{\\mathbf{R}} c^\\dag_{\\mathbf{R}}\\beta c_{\\mathbf{R}}. \n\\end{equation}\nHere, the translation operators in $\\mu=x,y,z$ directions are\n\\begin{equation}\n\\label{eq:Wlsn2}\nT_{\\mu}=\\sum_{\\mathbf{R}}c^\\dag_{\\mathbf{R}+\\mathbf{a}_{\\mu}}e^{i A_{\\mu}}(r\\beta-it\\alpha_{\\mu})c_{\\mathbf{R}},\n\\end{equation}\nwhere $t$ and $r$ are the hopping parameters with and without spin flip, respectively, $m$ is the mass characterizing the spin-orbit interaction, $\\mathbf{R}=(n_x,n_y,n_z)$ is the lattice coordinate, $\\mathbf{a}_{\\mu}$ is the unit lattice vector in $\\mu$ direction, and $\\alpha_{\\mu}$, $\\beta$ are the standard Dirac matrices:\n\\begin{equation*}\n  \\alpha_\\mu=\n  \\begin{pmatrix}\n    0&\\mathbf{\\sigma_\\mu}\\\\\n    \\mathbf{\\sigma_\\mu}&0\n  \\end{pmatrix},\\quad\\beta=\n  \\begin{pmatrix}\n    I&0\\\\\n    0&-I\n  \\end{pmatrix},\n\\end{equation*}\nwhere $\\sigma_{\\mu}$ are the Pauli matrices. \n\nTo take into account the effect of an external electric field and the interaction of electron spins on TI surfaces with magnetization, we introduce additional terms $H_E$ and $H_s$, respectively. Then, the total Hamiltonian is\n\\begin{eqnarray}\n \\label{eq:Ham}\n  H&=&H_0+H_E+H_s,\\\\\n  H_E&=&\\sum_{\\mathbf{R}}c^\\dag_{\\mathbf{R}}U(n_z)\n  \\begin{pmatrix}\n    I&0\\\\\n    0&I\n  \\end{pmatrix}\n  c_{\\mathbf{R}}\n\\end{eqnarray}\nwith the potential energy $U(n_z)=-E(n_z-L_z\/2)$, where $L_z$ is the TI film thickness. The surface spin contribution \\cite{Qi2008, Nomura2011, LiuPRL2009, Abanin2011}, see Fig.~\\ref{fig:EM}, is given by\n\\begin{equation}\nH_s=\\sum_{\\mathbf{R}}c^\\dag_{\\mathbf{R}} b(n_z) \\Sigma_s^+c_{\\mathbf{R}}, \\,\nb(n_z)=\n\\begin{cases}\nb_s,&n_z=L_z,\\\\\n-b_s,&n_z=0.\n\\end{cases}\n\\end{equation}\nHere, $\\Sigma_s^+=\\mathrm{diag}(\\mathbf{\\sigma}_z,\\mathbf{\\sigma}_z)$ and $b_s$ is the surface spin constant. \n    \n\\section{Results and Discussion} \nWe investigate the magnetoelectric effect by calculating the energy eigenvalues of electrons. Since our model has translational invariance in $x$ and $y$ directions, Eq.~(\\ref{eq:Ham}) can be written as \n\\begin{equation}\n H=\\sum_{k_x,k_y,n_z}c^{\\dag}_{n_z}(k_x,k_y){\\cal H}_{n_z,n_z'}(k_x,k_y)c_{n_z'}(k_x,k_y),\n\\end{equation}\nwhere ${\\cal H}_{n_z,n_z'}(k_x,k_y)$ is $4q(L_z+1)\\times 4q(L_z+1)$ matrix, see Appendix for details. By numerically performing an exact diagonalization of this Hamiltonian matrix, we solve the eigenvalue equation \n${\\cal H}(k_x,k_y)|\\lambda,k_x,k_y\\rangle=\\epsilon_{\\lambda}(k_x,k_y)|\\lambda,k_x,k_y\\rangle$.\nThe ME effect is described by the energy density of electrons:\n\\begin{equation}\n U_{\\rm tot}=\\frac{1}{\\mathcal{V}}\\sum_{k_x,k_y}{\\sum_{\\lambda}}'\\epsilon_{\\lambda}(k_x,k_y),\n\\end{equation}\nwhere $\\mathcal{V}$ is the TI volume and $\\sum_{\\lambda}'$ is the sum of eigenvalues below the Fermi level, which is fixed at $\\epsilon_F=0$ in the following.\nIn the presence of both electric $\\mathbf{E}$ and magnetic $\\mathbf{B}$ fields, $U_{\\rm tot}$ has a term $\\propto\\mathbf{E}\\cdot\\mathbf{B}$ as expected from the axion electrodynamics, Eq.~(\\ref{Eq:Enden}).\n\nThe energy density for a typical case is plotted in the inset of Fig.~\\ref{fig:thetaLz}. For fixed magnetic field, we change the electric field and plot the energy density as a function of $\\mathbf{E}\\cdot\\mathbf{B}$.\nFrom Eq.~(\\ref{Eq:Enden}), we estimate the parameter $\\theta$ by $\\theta=-4\\pi^2\\tan\\gamma$, where $\\tan\\gamma$ is the slope of the linear dependence in the inset of Fig.~\\ref{fig:thetaLz}. In all our calculations, we set the hopping parameters $t=r=1$, the mass $m=-1$, and choose the number of layers in $x$ direction $L_x$, so that the number of points in $k_x$ direction of the magnetic Brillouin zone is $L_x\/q=32$.\n\n\\begin{figure}\n\\includegraphics[width=0.97\\columnwidth]{thetaLzcom}\n\\caption{(Color online) Parameter $\\theta$ as a function of TI film thickness $L_z$. The inset shows the energy density of electrons in a weak electric field and its linear relationship to the product of electric and magnetic fields for $L_z=30$.}\n\\label{fig:thetaLz}\n\\end{figure}\n\nFirst, to verify our method, we study $\\theta$ as a function of TI thickness $L_z$ for the fixed magnetic field $B=\\pi\/5$ corresponding to $p=1$ and $q=10$, see the main panel of Fig.~\\ref{fig:thetaLz}. The data show that $\\theta$ is close to $\\pi$ at large $L_z$, whereas it decreases notably below $L_z\\approx10$. As $L_z$ approaches 1, the parameter $\\theta$ almost vanishes, which means that in very thin TI films the ME effect disappears. When the TI thickness is very small, it is expected that its surface wave functions hybridize with each other \\cite{Linder09,Essin2009}. To confirm it, we calculate the spacial profiles of the wave functions in $z$ direction for the surface states.  At the $\\Gamma$ point $(k_x=k_y=0)$, they are calculated for several values of $L_z$ and shown in Fig.~\\ref{fig:LzComb}. We observe that for $L_z=30$ the wave functions are strongly localized near the TI surfaces, whereas overlap of these wave functions increases with decreasing TI thickness. At $L_z=4$ the top and bottom wave functions almost completely overlap. This clearly indicates that the reduction of the ME effect is associated with the hybridization of the surface wave functions. From Fig.~\\ref{fig:thetaLz}, the value of $L_z$ below which the ME effect starts to significantly diminish can be estimated as $\\sim 10$. For typical TIs \\cite{Cui2004,Hsieh2008,Chen2009,Xia2009,Hsieh2009}, this value corresponds to $\\sim 10$ nm. \n\n\\begin{figure}\n\\centering \n\\includegraphics[width=1\\columnwidth]{hybridization_combined}\n\\caption{(Color online) The hybridization of surface wave functions $\\Psi$ for TI thicknesses $L_z=4,7,10$, and $30$. The red dashed line corresponds to the bottom TI surface wave function and the blue solid line corresponds to the top surface wave function.}\n\\label{fig:LzComb}\n\\end{figure}\n\nSo far, we have considered a TI in a weak electric field. Experimentally, however, ultrahigh electric fields can be achieved by ionic liquid gating \\cite{Checkelsky2012}, and it is therefore interesting to study a nontrivial response of a TI to both strong electric and magnetic fields.  In the presence of a strong magnetic field, Landau levels form on top and bottom TI surfaces. The existence of well-defined Landau levels in TIs has been confirmed by recent experimental observations of the quantum Hall effect \\cite{Brune2011,Chang2013}. In our simulation, we fix for simplicity the Landau levels on the bottom surface, whereas the gating induced electric field, $E_g$, shifts the energy levels on the top surface. As the gate voltage changes from $V_g\\equiv E_gL_z=0$, the Landau level energies on the top surface move. Figure~\\ref{fig:EgLz}(a) shows that Landau levels cross the Fermi level\nat the following gate voltages: $V_{-5},\\,V_{-4},\\,V_{-3},\\ldots,V_{5}$. In the nonlinear regime of strong electric or magnetic fields, the axion electrodynamics corresponding to $\\theta=\\pm\\pi$ in general is not expected to be applicable. To probe this regime, we vary the electric field by small $\\delta \\mathbf{E}$ near $\\mathbf{E}_g$, i. e. the total electric field is $\\mathbf{E}=\\mathbf{E}_g+\\delta\\mathbf{E}$, and study the response to $\\delta\\mathbf{E}\\cdot\\mathbf{B}$. \nThen $\\delta\\mathbf{E}$ has a role of $\\mathbf{E}$ in Eq.~(\\ref{Eq:Enden}) and $\\theta(E_g,B)$ is defined as the overall coefficient describing the ME effect response.\n\nFigure~\\ref{fig:EgLz}(b) shows $\\theta\/\\pi$ as a function of $V_g$. One can see that there are several plateaus where $\\theta\/\\pi$ is approximately quantized in integers $\\{-4,-3,-2,-1,0,1,2,3,4,5\\}$. Every new quantization plateau in Fig.~\\ref{fig:EgLz}(b) occurs at the points where Landau levels cross the Fermi level.\nThe exact quantization would be in agreement with the prediction of Ref.~\\onlinecite{Tse2010} for weak electric fields. \nAs we show below, the deviation from the exact quantization is mostly attributed to the asymmetry of TI surface wave functions due to the application of the large gate voltage. Thus, Fig.~\\ref{fig:EgLz}(b) shows an important fact that although time-reversal symmetry is broken in a 3D TI by a strong magnetic field, $\\theta$ is not arbitrary as it has been claimed in Refs.~\\onlinecite{Coh2011,Sitte2012}, but rather quantized as long as the Fermi level is still within the bulk gap. In accord with this, the two continuous regions of the dependence in Fig.~\\ref{fig:EgLz}(b) at the left and right edges correspond to the cases when the Fermi level enters continuous (valence or conduction) bands.  In the two continuous regions, however, our results only qualitatively show the behavior of the parameter $\\theta$, because we do not consider any screening effects coming from conduction and valence bands that may suppress the $\\theta$-term, and also because beyond the range bounded by $V_{-5}$ and $V_{5}$, see Fig.~\\ref{fig:EgLz}(a), there are multiple crossings of Landau levels very close to each other. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{thetacom}\n\\caption{(Color online) (a) The TI energy levels (at $\\Gamma$ point) as a function of gate voltage $V_g$ measured in units of $E_{g}L_z$. The surface Landau levels cross the Fermi level. The red lines represent the surface Landau levels and the blue line is the Fermi level. (b) The ME response quantization in a TI vs $V_g$. }\n\\label{fig:EgLz}\n\\end{figure}\n\nTo investigate the hypothesis that the lack of exact quantization in $\\theta$ is attributed to the asymmetry of TI surface wave functions due to gating, we repeat the same calculation for a thicker TI film. The details of the quantization of $\\theta$ as a function of gate voltage $V_g$ are shown in Fig.~\\ref{fig:Lz10vsLz20} for two different TI film thicknesses. The results for $L_z=10$ are shown by blue circles and for $L_z=20$ by red triangles. At a fixed gate voltage $V_g$ ($=E_{g}L_{z}$), applied electric field for $L_z=10$ case is two times stronger than that for $L_z=20$. The comparison shows that when $L_z$ increases two times the slope of each quantization plateau becomes two times smaller. This indicates that for thick enough TI films the plateaus should become horizontal, because in this case the asymmetry of surface wave functions can be disregarded since their overlap is too small. As a result, for the thick films, $\\theta$ depends only on the surface states, but not the bulk states. When the bulk gap is not destroyed, $\\theta$ is determined by the surfaces and thus quantized as the Hall conductivity is quantized, i.e.,\n\\begin{equation}\n\\frac{\\theta}{\\pi} = \\nu_t -\\nu_b,\n\\end{equation}\nwhere $\\nu_{t(b)}$ are the Landau level filling factors for the top (bottom) TI surfaces. In the case of zero applied gate voltage,  $\\nu_{t}=-1\/2$, $\\nu_{b}=1\/2$ and therefore $|\\theta\/\\pi|=1$, c.f. Fig.~\\ref{fig:thetaLz}. \n\nBy applying a magnetic field to the TI model, Eq.~(\\ref{eq:Wlsn}), we break time-reversal symmetry and as a result the system crosses from symmetry class AII, where $\\theta$ is quantized $0$ or $\\pi$, to class A where in general $\\theta$ can be arbitrary. Thus we show that depending on the perturbations (orbital or Zeeman magnetic fields or electric field), $\\theta$ is still quantized in class A for our model. The quantization rules can be rather nontrivial depending on a combination of perturbations, however, we can successfully explain these rules based on the Landau level physics.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=0.97\\columnwidth]{thetaLz10vsLz20}\n\\caption{(Color online) The quantization of $\\theta$ as a function of gate voltage $V_g$ for $L_z=10$ (blue circles) and $L_z=20$ (red triangles). The comparison shows that when the TI thickness $L_z$ increases two times the slope of each quantization plateau becomes two times smaller.}\n\\label{fig:Lz10vsLz20}\n\\end{figure}\n\nFinally, we study the effects of bulk magnetic interaction on the ME effect. The Zeeman type interaction between the electron spin and applied magnetic field is described by \n\\begin{eqnarray}\n\\label{minus}\n\\mathcal{H}'&=&\\mu_BB_z\n\\begin{pmatrix}\ng_1\\sigma_z&0\\\\\n0&g_2\\sigma_z\n\\end{pmatrix},\n\\end{eqnarray}\nwhere $g_1$ and $g_2$ are the effective $g$ factors depending on the TI materials. In most of cases, for materials such as Bi$_2$Se$_3$ and Bi$_2$Te$_3$, $g_1$ and $g_2$ have opposite signs \\cite{Liu2010}. Therefore, as a simplified model, we consider the case of $g_2=-g_1$. In addition, we study the influence of the exchange interaction between the electron spins and magnetic impurities that can be introduced by the appropriate doping of the TI bulk. Within the virtual crystal approximation, this interaction has the form\n\\begin{eqnarray}\n\\label{plus}\n\\mathcal{H}'&=&J\\overline{M}_z\n\\begin{pmatrix}\n\\sigma_z&0\\\\\n0&\\sigma_z\n\\end{pmatrix},\n\\end{eqnarray}\nwhere $J$ is the exchange interaction constant and $\\overline{M}_z$ is the mean value of magnetic moments in the bulk. These two interactions, Eqs.~(\\ref{minus}) and (\\ref{plus}), can be covered by two simple model Hamiltonians: $\\mathcal{H}'=b_b\\Sigma_z^{\\pm}$ with $\\Sigma_z^{\\pm}=\\mathrm{diag}(\\mathbf{\\sigma}_z, \\pm\\mathbf{\\sigma}_z)$ and $b_b$ being the bulk spin constant. The sign ``$+$'' in $\\Sigma_z^+$ indicates bulk electrons coupling with magnetization, while ``$-$'' sign indicates coupling with the external magnetic field.  \n\nThe calculation results for magnetic field $B=\\pi\/10$ are shown in Fig.~\\ref{fig:bulkspin}, where the red and blue curves correspond to $\\Sigma_z^-=\\mathrm{diag}(\\mathbf{\\sigma}_z,-\\mathbf{\\sigma}_z)$, red is for $L_z=15$ and blue is for $L_z=10$.  The black and green curves correspond to $\\mathrm{diag}(\\mathbf{\\sigma}_z,\\mathbf{\\sigma}_z)$ case, with the black for $L_z=15$ and green for $L_z=10$. One can see from Fig.~\\ref{fig:bulkspin} that $\\theta\/\\pi=1$ persists approximately up to $b_b=1$, which corresponds to the TI regime. At $b_b=|m|=1$, the system crosses to the Weyl semimetal regime \\cite{Wan2011,Burkov2011} for $\\mathrm{diag}(\\mathbf{\\sigma}_z,\\mathbf{\\sigma}_z)$ case and therefore no quantization of $\\theta$ beyond $b_b=1$ is expected \\cite{Zyuzin2012}. These results show remarkable robustness of the ME effect in TI phase even for very strong magnetic fields or high concentration of magnetic impurities that affect the bulk structure of the TI. It seems to be irrespective of microscopic details of coupling with the magnetic impurities or field, since it works equally well in the TI regime for the toy models with both $\\pm$ signs.\n \n\\begin{figure}\n\\centering \n\\includegraphics[width=0.97\\columnwidth]{bulkspin}\n\\caption{(Color online) The bulk spin interaction influence on the ME effect. The triangles and circles represent $\\Sigma_z^+=\\mathrm{diag}(\\mathbf{\\sigma}_z,\\mathbf{\\sigma}_z)$ case, whereas ``$\\times$'' and ``$+$'' symbols represent $\\Sigma_z^-=\\mathrm{diag}(\\mathbf{\\sigma}_z,-\\mathbf{\\sigma}_z)$ case.}\n\\label{fig:bulkspin}\n\\end{figure}\n\n\\section{Summary}\n\nWe have shown that the ME effect in TIs survives even in both strong magnetic and electric fields even though time-reversal symmetry is broken. We have described the ME effect  beyond the linear response regime and have confirmed that it is robust but takes a fascinating form of multiple quantization related to Landau levels crossing of the Fermi level. In addition, the influence of the TI surface hybridization has been studied in very thin TI films, describing how the ME effect vanishes in the limit of very thin TIs.  Furthermore, the ME effect is shown to persist even for strong bulk interactions with magnetic field or magnetic impurities.  \n\n\\acknowledgments \n\nWe would like to thank the staff of the Center for Computational Materials Science at the IMR, Tohoku University for their support of the supercomputing facilities. K.N. was supported by Grant-in-Aid for Scientific Research (Nos. 24740211 and 25103703) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. O.A.T. acknowledges support  by the Grants-in-Aid for Scientific Research (Nos. 25800184 and 25247056) from the MEXT, Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Nodal components of random functions}\n\nGiven a ``nice\" function\n$f:{\\mathbb R}^{d}\\rightarrow{\\mathbb R}$, $d\\ge 2$, or $f:\\M\\rightarrow{\\mathbb R}$ with $\\M$ a compact Riemannian $d$-manifold, the nodal set\nof $f$ is its zero set $f^{-1}(0)$; if $f$ is Morse, then its nodal set is a smooth hypersurface in both the ``Euclidean\" and\n``Riemannian\" contexts.\nIn either scenario, the {\\em nodal components} of $f$ are the connected components of the nodal set, and\nthe {\\em nodal domains} are the (positive or negative) connected components of the {\\em complement} of the nodal set.\nThe most basic question one is interested is in the {\\em nodal count} of $f$, i.e.\nthe total number of the nodal components of $f$, also yielding the total number of nodal domains of $f$ via Euler's identity,\nat least, in the Riemannian context $f:\\M\\rightarrow{\\mathbb R}$ (where the nodal set is of finite\nhypersurface volume, thanks to the assumed compactness of $\\M$).\n\nTo make sense of the analogous question in the Euclidean context $f:{\\mathbb R}^{d}\\rightarrow{\\mathbb R}$, one usually takes a large parameter $R>0$\nand studies the asymptotics of the nodal count of the restriction $f|_{B(R)}$ to the centred radius-$R$ ball $B(R)\\subseteq {\\mathbb R}^{d}$, as\n$R\\rightarrow\\infty$. Other than the nodal count, one might refine the said question by separately counting the nodal components of $f$ belonging to\ne.g. a given diffeomorphism type or otherwise, study the mutual positions of the components (``nestings\") etc.\n\n\\vspace{2mm}\n\nUnderstanding the ``typical\" nature of the nodal structures of {\\em Gaussian random fields}, rather than individual functions,\nis an actively pursued subject within several disciplines, in the last few years in particular.\nLet $F:{\\mathbb R}^{2}\\rightarrow {\\mathbb R}$ (more generally, $F:{\\mathbb R}^{d}\\rightarrow{\\mathbb R}$) be a Gaussian random field, that will be assumed {\\em stationary},\nand $R>0$ be a large parameter.\nThen the number $\\pazocal{N}(F,R)$ of nodal components of $F$ {\\em fully contained} inside $B(R)$ is a random variable; alternatively, one\ncould count those components merely {\\em intersecting} $B(R)$. It was shown by Nazarov-Sodin ~\\cite{NS09,So12,NS15} that, under very\nmild smoothness and non-degeneracy assumptions on the law of $F$, there exists a constant $c_{NS}\\ge 0$ so that\n$\\frac{\\pazocal{N}(F,R)}{\\operatorname{Vol}(B(R))}$ converges to $c_{NS}$ both a.s. and in mean, i.e.\n\\begin{equation}\n\\label{eq:cNS conv mean}\n{\\mathbf E}\\left[\\left| \\frac{\\pazocal{N}(F,R)}{\\operatorname{Vol}(B(R))} - c_{NS}  \\right|\\right] \\rightarrow 0\n\\end{equation}\nas $R\\rightarrow\\infty$. Sarnak-Wigman ~\\cite{SW18}, and Beliaev-Wigman ~\\cite{BW17} further developed the techniques\ndue to Nazarov-Sodin by considering the more refined questions of separate nodal counts belonging to particular diffeomorphism types or given hypersurface measure.\n\n\n\\subsection{Random band-limited functions on smooth manifolds} \\label{s:band_lim}\n\nRather than for its own intrinsic interest, this Euclidean scenario above serves as a ground state for the Riemannian one, namely,\nas its scaling limit. Here we consider\na sequence (``ensemble\") $\\{f_{l}\\}_{l\\in\\Lc}$, of smooth random Gaussian fields $f_{l}:\\M\\rightarrow{\\mathbb R}$, satisfying a natural scaling\nproperty, with the scaling parameter $l$ lying in some countable set $\\Lc$, and our objective is to study the distribution\nof the {\\em total} number of nodal components of $f_{l}$ as $l\\rightarrow\\infty$, their typical topology, geometry, relative positions,\nand other important properties. A particularly important such ensemble,\nmotivating the work ~\\cite{SW18}, is\nthe ensemble of band-limited functions, depending on a fixed number $\\alpha\\in [0,1]$. This includes the important ensembles of\nrandom degree-$l$ spherical harmonics (see \\S\\ref{sec:spher harm RMW} below), and Arithmetic Random Waves (\\S\\ref{sec:ARW} below).\n\n\nIt is well-known that, since we assumed $\\M$ to be a smooth, compact, Riemannian $d$-manifold, the space $L^{2}(\\M)$ of square-summable functions on $\\M$\nhas an orthonormal basis $\\{\\varphi_{j}\\}_{j=1}^{\\infty}$ consisting of Laplace eigenfunctions, i.e.\n\\begin{equation}\n\\label{eq:Helmholts}\n\\Delta \\phi_{j}+t_{j}^{2}\\phi_{j} = 0,\n\\end{equation}\nwhere $\\Delta$ is the Laplace-Beltrami operator on $\\M$ acting on $L^{2}$, and $\\{t_{j}\\}_{j\\ge 0}$ is its purely discrete spectrum\n$$0=t_{0}\\le t_{2}\\le\\ldots,$$ satisfying $t_{j}\\rightarrow\\infty$. For a ``band\" $\\alpha\\in [0,1)$ and spectral\nparameter $T>0$ (with the intention of taking the limit $T\\rightarrow\\infty$), we define ~\\cite{SW18} the random\nband-limited functions to be\n\\begin{equation}\n\\label{eq:fT band lim alpha<1}\nf_{T}(x)=f_{\\alpha;T}(x)=\\frac{1}{\\left|\\{j:\\alpha\\cdot T<t_{j}<T\\}\\right|^{1\/2}}\\sum\\limits_{\\alpha\\cdot T<t_{j}<T}c_{j}\\varphi_{j}(x),\n\\end{equation}\nwhere the $c_{j}$ are i.i.d. standard Gaussian. For $\\alpha=1$ the above definition usually makes no sense, as the summation on the r.h.s.\nof \\eqref{eq:fT band lim alpha<1} is typically one summand (or empty), so in this case we interpret the summation as\n\\begin{equation}\n\\label{eq:fT band lim alpha=1}\nf_{1;T}(x)=\\frac{1}{\\left|\\{j:T-\\eta(T)<t_{j}<T\\}\\right|^{1\/2}}\\sum\\limits_{T-\\eta(T)<t_{j}<T}c_{j}\\varphi_{j}(x),\n\\end{equation}\nwith the convention $\\eta(T)=o_{T\\rightarrow\\infty}(T)$ but $\\eta(T)\\rightarrow\\infty$.\n\nAlternatively, $f_{T}(\\cdot)$ is the (uniquely defined) centred Gaussian random field of covariance\n\\begin{equation}\n\\label{eq:KT covar func}\nK_{T}(x,y)=K_{\\alpha;T}:={\\mathbf E}[f_{\\alpha;T}(x)\\cdot f_{\\alpha;T}(y)] = \\sum \\phi_{j}(x)\\cdot\\phi_{j}(y),\n\\end{equation}\n$x,y\\in\\M$, where the summation is over the same energy window as \\eqref{eq:fT band lim alpha<1} for $\\alpha<1$\n(resp. \\eqref{eq:fT band lim alpha=1} for $\\alpha=1$); we identify $K_{T}(\\cdot,\\cdot)$ as the {\\em spectral projector}\nfor the corresponding energy window. The kernel $K_{T}(\\cdot, \\cdot)$ and its derivatives possess scaling limits once appropriately scaled by $T$,\nsee \\S\\ref{sec:proofs outline} for more details, and in particular \\eqref{eq:covar func band lim der conv}.\n\n\\subsubsection{Random spherical harmonics and Berry's Random Wave Model}\n\\label{sec:spher harm RMW}\n\nIt is well-known that the Laplace eigenfunctions on the $2$-sphere $S^{2}\\subseteq {\\mathbb R}^{3}$, that is the spherical harmonics, are restrictions of harmonic polynomials of some degree $l\\ge 1$. The space of degree-$l$ spherical harmonics is of\ndimension $2l+1$, so given a number $l$ we may obtain an $L^{2}$-orthonormal basis $\\Phi_{l}:=\\{\\eta_{l;1},\\ldots,\\eta_{l;2l+1}\\}$,\nand define\n\\begin{equation}\n\\label{eq:Tl spher harm}\nT_{l}(x):=\\frac{\\sqrt{4\\pi}}{\\sqrt{2l+1}}\\sum\\limits_{m=1}^{2l+1}a_{m}\\eta_{l;m}(x)\n\\end{equation}\nwith $\\{a_{m}\\}_{m=1}^{2l+1}$ standard Gaussian i.i.d.; the law of the random spherical harmonics $T_{l}(\\cdot)$ is\ninvariant w.r.t. the choice of $\\Phi_{l}$. The random fields $T_{l}(\\cdot)$ are the Fourier components of every\nrotation invariant random field on $S^{2}$, hence its importance in a variety of disciplines within mathematics,\nphysics and cosmology.\n\n\nEquivalently, the random spherical harmonic $T_{l}(\\cdot)$ is a centred\nGaussian random field defined via the covariance function $$r_{l}(x,y)=P_{l}(\\cos d(x,y)),$$ $x,y\\inS^{2}$,\n$d(\\cdot,\\cdot)$ is the spherical geodesic distance, and $P_{l}(\\cdot)$ is the degree-$l$ Legendre polynomial.\nSince, by the standard Hilb's asymptotics ~\\cite{Szego}, $$P_{l}(\\cos\\theta)\\approx J_{0}((l+1)\\theta),$$\nthe geometry of the nodal line of $T_{l}$ could be compared to the geometry of the nodal line of the stationary isotropic\nrandom field on ${\\mathbb R}^{2}$ defined by the covariance function $J_{0}(\\|x\\|)$, usually referred to ``Berry's Random Wave Model\"\n(RWM), for it is postulated ~\\cite{Berry 1977} to be a stand-in for deterministic Laplace eigenfunctions on generic chaotic surfaces.\n\n\\subsubsection{Arithmetic random waves}\n\\label{sec:ARW}\n\nThe Arithmetic Random Waves are random Gaussian toral Laplace eigenfunctions $\\{f_{n}:\\Tb^{2}\\rightarrow{\\mathbb R}\\}_{n\\in S}$,\nwhere $\\Tb^{2}={\\mathbb R}^{2}\/{\\mathbb Z}^{2}$ is the $2$-dimensional standard torus, and $S=\\{a^{2}+b^{2}:\\:a,b\\in{\\mathbb Z}\\}$ is the set of all numbers\nexpressible as sum of two integer squares. For $n\\in S$ let $$\\Lambda_{n}:=\\{(a,b)\\in {\\mathbb Z}^{2}:\\: a^{2}+b^{2}=n\\}$$ be the set\nof all lattice points lying on the centred circle of radius $\\sqrt{n}$. We may define\n\\begin{equation}\n\\label{eq:fn ARW def}\nf_{n}(x)=\\sum\\limits_{\\lambda\\in\\Lambda_{n}} a_{\\lambda}e(\\langle \\lambda,x  \\rangle),\n\\end{equation}\nwhere $e(t)=e^{2\\pi i t}$, and the $\\{a_{\\lambda}\\}_{\\lambda\\in\\Lambda_{n}}$ are complex standard Gaussian i.i.d., save to the condition\n$a_{-\\lambda}=\\overline{a_{\\lambda}}$, so that $f_{n}$ are real valued.\n\nThe random fields $f_{n}$ are the ``Arithmetic Random Waves\"\nthat serve as a motivation to our research, following the work ~\\cite{RudWig2018}; $f_{n}$ should be compared\nto the band-limited functions \\eqref{eq:fT band lim alpha=1}, with $n\\approx T^{2}$\n(and, thanks to the spectral multiplicity, we may take an infinitesimal energy window).\nEquivalently to the explicit definition \\eqref{eq:fn ARW def},\n$f_{n}$ could be defined as the stationary centred Gaussian random field on $\\Tb^{2}$ with the covariance function\n\\begin{equation}\n\\label{eq:ARW covar def}\nr_{n}(x)={\\mathbf E}[f_{n}(y)\\cdot f_{n}(y+x)] = \\frac{1}{|\\Lambda_{n}|}\\sum\\limits_{\\lambda\\in\\Lambda_{n}}\ne(\\langle \\lambda,x \\rangle),\n\\end{equation}\nor via the spectral measure of $f_{n}$, i.e. the atomic measure\n\\begin{equation}\n\\label{eq:ARW spec measure rho}\n\\rho_{n}:=\\frac{1}{|\\Lambda_{n}|}\\sum\\limits_{\\lambda\\in\\Lambda_{n}}\\delta_{\\lambda\/\\sqrt{n}}\n\\end{equation}\non $S^{1}\\subseteq{\\mathbb R}^{2}$.\n\n\\subsection{Direction distribution and statement of the principal result}\n\n\\subsubsection{Direction distribution on $\\mathbb{T}^d$}\n\nIn $2$d Rudnick and Wigman ~\\cite{RudWig2018} proposed to study the {\\em ``direction distribution\"} of the nodal line, a different quantity they\nintroduced for functions defined on the $2$d standard torus $\\Tb^{2}={\\mathbb R}^{2}\/{\\mathbb Z}^{2}$, also related to ~\\cite{Swerling}.\nLet $\\zeta\\inS^{1}$ be\na direction, and $f:\\Tb^{2}\\rightarrow{\\mathbb R}$ any ``nice\" function. The number\n\\begin{equation}\n\\label{eq:Nzeta torus}\n\\mathcal{N}_{\\zeta}(f):= \\left|\\left\\{x\\in \\Tb^{2}:f(x)=0,\\, \\frac{\\nabla f(x)}{\\|\\nabla f(x)\\|}=\\pm \\zeta  \\right\\} \\right|\n\\end{equation}\nis the number of\npoints on the nodal line of $f$ normal to $\\zeta$; equivalently, $\\mathcal{N}_{\\zeta}(f)$ is the number of nodal points of $f$ in direction\n$\\pm\\xi:=\\zeta^{\\perp}$. The direction distribution carries a lot of information on the nodal line of $f$; for example, the number of\nnodal components of $f$ on $\\Tb^{2}$ essentially majorizes ~\\cite{KuWiAdv} the nodal count of $f$ for every $\\zeta$,\nas every nodal component of $f$ of trivial\nhomology contains at least two points tangent to $\\xi$, and it is usually easy to control the contribution of all the other\ncomponents.\n\nFor the $\\mathcal{N}_{\\zeta}(\\cdot)$ corresponding to\ntoral Laplace eigenfunctions Rudnick and Wigman ~\\cite{RudWig2018} gave the optimal upper bounds, as well as evaluated their total expected number for the associated Gaussian random model, ``Arithmetic Random Waves\" \\eqref{eq:fn ARW def}, precisely, appealing to the Kac-Rice method. Their result, however, does not allow for the separation of the $\\xi$-tangencies into nodal components of $f$, i.e. how many points on a\n{\\em given component} represent $\\xi$, or how many nodal components of $f$ represent the direction $\\xi$ {\\em precisely} $k$ times.\n\n\\subsubsection{Direction distribution on curved surfaces}\n\nThe purpose of this manuscript is two-fold. First, we propose, what it seems, a natural generalisation of the direction distribution concept for smooth manifolds with curvature, and study its properties for the random band-limited functions \\eqref{eq:fT band lim alpha<1} (and\n\\eqref{eq:fT band lim alpha=1}).\nSecond, in this random setting (random band-limited functions, including, in particular Arithmetic Random Waves) we aim at refining\nthe said results due to Rudnick-Wigman by keeping separate accounts for the number of $\\xi$-tangencies on individual components rather than merely the total number of $\\xi$-tangencies (though the latter is useful for our purposes).\n\nLet us now introduce a space of vector fields that play a central role in our article. We let $\\mathcal{V}(\\M)$ be the class of all $C^{\\infty}$-smooth vector fields on $\\M$, with finitely many zeros; the class $\\mathcal{V}(\\M)$ is non-empty for\nevery smooth $\\M$ by Lemma \\ref{l:vfs_finite_zeroes}. First, given $f:\\M\\rightarrow{\\mathbb R}$ a nice function, and $V\\in\\mathcal{V}(\\M)$, we introduce\n\\begin{equation*}\n\\mathcal{N}_{V}(f)= \\#\\{x:\\:V(x)\\ne 0,\\, f(x)=Vf(x)=0\\},\n\\end{equation*}\nthe announced generalisation of the ``flat\" direction distribution, with the variable direction $V(x)$ in place of\n$\\xi=\\zeta^{\\perp}$ in \\eqref{eq:Nzeta torus}. Other than for the torus, a precise asymptotic expression for the total number of\n$V$-tangencies for the random spherical harmonics \\eqref{eq:Tl spher harm} was recently obtained \\cite{E18} by the first author.  Related results were obtained by Dang and Rivi\\`ere \\cite{DaRiJEMS} following some substantial and general results of Gayet and Welschinger \\cite{GaWeJussieu}.\n\n\n\nNow we separate the tangency counts for individual nodal components of $f$: for a nodal component $\\gamma\\subseteq f^{-1}(0)$ define\nthe number of tangencies w.r.t. $V$,\n\\begin{equation*}\nT_{V}(\\gamma)= \\#\\{x\\in \\gamma:\\: V(x)\\ne 0,\\, f(x)=Vf(x)=0\\},\n\\end{equation*}\nand\n\\begin{equation*}\n\\pazocal{N}_{V}(f,k) :=\\#\\{\\gamma  \\subseteq f^{-1}(0):\\: T_{V}(\\gamma)=k\\}\n\\end{equation*}\nis the total number of nodal components of $f$ with precisely $k$ tangencies w.r.t. $V$,\nso that we have the\nidentity\n\\begin{equation}\n\\label{eq:tot numb tang=sum mean}\n\\mathcal{N}_{V}(f) = \\sum\\limits_{k=1}^{\\infty}k\\cdot \\pazocal{N}_{V}(f,k).\n\\end{equation}\nFinally, we let\n\\begin{equation*}\n\\pazocal{N} (f) = \\sum\\limits_{k=0}^{\\infty}\\pazocal{N}_{V}(f,k)\n\\end{equation*}\nbe the total number of the nodal components of $f$.\n\n\\subsubsection{Statement of the principal result}\n\nOur main result concerns the asymptotic law for the direction distribution measure corresponding to the band-limited functions\n$f_{\\alpha;T}(\\cdot)$ in \\eqref{eq:fT band lim alpha<1}-\\eqref{eq:fT band lim alpha=1}.\nFollowing the approach of Sarnak-Wigman ~\\cite{SW18},\nwe may incorporate, or ``encapsulate\", all the individual counts $\\pazocal{N}_{V}(f,\\cdot)$ into a single (random) probability measure,\nthe ``direction distribution measure\",\n\\begin{equation}\n\\label{eq:mu(f) ddm def}\n\\mu_{f}(V) =\\frac{1}{\\pazocal{N}(f)}\\sum\\limits_{\\gamma\\subseteq f^{-1}(0)}\\delta_{T_{V}(\\gamma)} =\\frac{1}{\\pazocal{N}(f)}\\sum\\limits_{k=0}^{\\infty}\\pazocal{N}_{V}(f,k)\\cdot\\delta_{k},\n\\end{equation}\non ${\\mathbb Z}_{\\ge 0}$.\nGiven two probability measures $\\mu_{1},\\mu_{2}$ on ${\\mathbb Z}$ we will use the total variation distance function\n\\begin{equation}\n\\label{eq:D tot var dist def}\n\\mathcal{D}(\\mu_{1},\\mu_{2}) = \\sup\\limits_{F\\subseteq {\\mathbb Z}_{\\ge 0}} |\\mu_{1}(F)-\\mu_{2}(F)|.\n\\end{equation}\n\n\n\\begin{theo}\n\\label{thm:main meas conv}\n\nGiven $\\alpha\\in [0,1]$, there exists a (deterministic) probability measure $\\mu_{\\alpha}$ on ${\\mathbb Z}_{\\ge 0}$,\nsupported on the positive even integers $2{\\mathbb Z}_{>0}$, so that for all $V\\in\\mathcal{V} (\\M)$ and every $\\epsilon>0$,\n\\begin{equation}\n\\label{eq:main meas conv}\n\\lim\\limits_{T\\rightarrow\\infty}\\pazocal{P}r\\left(\\mathcal{D}(\\mu_{f_{\\alpha;T}}(V),\\mu_{\\alpha})>\\epsilon\\right)=0,\n\\end{equation}\nwhere $\\mathcal{D}(\\cdot,\\cdot)$ is the total variation distance \\eqref{eq:D tot var dist def}.\n\n\\end{theo}\n\n\\subsubsection{On the principal result}\n\nThe band-limited functions \\eqref{eq:fT band lim alpha<1}-\\eqref{eq:fT band lim alpha=1} is a particular case of an aforementioned ensemble of Gaussian random fields possessing a natural scaling by the wave number $T$, see \\S\\ref{sec:proofs discussions} below, and, in particular \\eqref{eq:covar func band lim der conv}.\nOur proofs are applicable for such ensembles with scaling,\nin a more general scenario than merely the band-limited functions, as long as an analogue of \\eqref{eq:covar func band lim der conv} holds,\nwith $T$ replaced by the scaling parameter (see e.g. the important application on Arithmetic Random Waves from \\S\\ref{sec:ARW},\nin \\S\\ref{sec:appl ARW} below).\nIn general, in a scenario like \\eqref{eq:main meas conv}, when probability measures (on ${\\mathbb Z}_{\\ge 0}$) weak-$*$\nconverge to a limit measure $\\mu_{\\alpha}$, it forces the limit measure to have $\\mu_{\\alpha}({\\mathbb Z}_{\\ge 0}) \\le 1$\nby Fatou's Lemma, and $\\mu_{\\alpha}({\\mathbb Z}_{\\ge 0}) < 1$ would mean escape of probability to infinity. As part\nof Theorem \\ref{thm:main meas conv}, we will be able to rule this kind of losses of mass, cf. \\cite[Theorem 1.1]{SW18}.\n\nWe stress that, despite the fact that the measures $\\mu_{f_{\\alpha;T}}$ are {\\em random}, the limit measure $\\mu_{\\alpha}$\nis {\\em deterministic}, and, notably, in addition, $\\mu_{\\alpha}$ is {\\em independent} of $V$.\nThanks to the asymptotic scaling \\eqref{eq:covar func band lim der conv}, we will be able to compare the nodal counts $\\pazocal{N}_{V}(f,k)$ to\nthe nodal counts of some Gaussian isotropic random field $\\mathfrak{g}_{\\alpha}(\\cdot)$ on ${\\mathbb R}^{2}$\n(to be defined in \\S\\ref{sec:proofs outline}),\nwith $V\\equiv \\xi$ a constant vector field,\nup to an admissible error term. The measures $\\mu_{\\alpha}$ are $V$-independent, since, by the isotropic property of $\\mathfrak{g}_{\\alpha}(\\cdot)$,\nthe corresponding distribution of $\\xi$-tangencies counts is independent of $\\xi$.\nThe support of $\\mu_{\\alpha}$ is a manifestation of the fact that the number of $\\xi$-tangencies\nfor a simple smooth planar curve is necessarily {\\em even}, unless a degeneracy occurs, with probability $0$ (though,\nstrictly speaking, it is possible to construct simple curve with an odd number of $\\xi$-tangencies). However,\nfor $k>0$ even it is possible to construct a simple curve with number of $\\xi$-tangencies being precisely $k$, that occurs\nas a nodal component of the scaled random field with positive density (the most subtle or delicate case being $\\alpha=1$, cf.\n~\\cite[Proposition $5.3$]{SW18} and ~\\cite{CanSar18}).\nWe refer the reader to \\S\\ref{sec:proofs outline} for more details on proofs and intuitions as of why these peculiarities hold.\n\n\\subsubsection{$T^*\\M$ and conormal cycles}\nWe conclude this section by emphasizing that the microlocal geometry of nodal sets, or more specifically the interaction between ``cotangent\/conormal spaces\" to nodal sets and various other geometric quantities of these submanifolds, is a natural topic of study. The work of Dang and Rivi\\`ere gives asymptotics pertaining to the equidistribution (in $T^* \\M$) of ``conormal cycles\" for $f_T$ on general compact manifolds \\cite{DaRiJEMS}, with conormal cycles being phase-space quantities related to the conormal bundle of $f^{-1}_T(0)$, namely $N^*(\\{ f_T = 0 \\}) \\subset T^* \\M$.  In other words, for example in odd dimensions, the authors show that the expected value of a natural and re-scaled volume measure on the conormal bundle to the nodal set $N^*(\\{ f_T = 0 \\})$ converges to a uniform volume measure on $T^*\\M$ as $T \\rightarrow \\infty$.\nIt is worth noting that Dang-Rivi\\`ere were themselves motivated by the work \\cite{GaWeJussieu} on expected Betti numbers for nodal sets associated to elliptic pseudodifferential operators; a local refinement of their lower bound was very recently obtained by Wigman \\cite{Wig19}.\n\n\n\\subsection{Applications to arithmetic random waves}\n\n\\label{sec:appl ARW}\n\n\nRecall the Arithmetic Random Waves $f_{n}$ in \\eqref{eq:fn ARW def} for $n\\in S$, the set of numbers expressible as sum of two squares, the corresponding covariance function \\eqref{eq:ARW covar def},\nand their spectral measure \\eqref{eq:ARW spec measure rho}.\nIt is well-known ~\\cite{KaKo,ErHa},\nthat for a ``generic\" (density-$1$) sequence $\\{n\\}\\subseteq S$, the measures $\\rho_{n}$ equidistribute\non $S^{1}$, i.e.\n\\begin{equation}\n\\label{eq:rhon equidistr}\n\\rho_{n}\\Rightarrow \\frac{d\\theta}{2\\pi}.\n\\end{equation}\nHowever, there exist ~\\cite{Cil,KuWiMAnn} other weak-$*$ partial limits of the sequence $\\{\\rho_{n}\\}_{n\\in S}$.\nIt was established ~\\cite{KKW,So12,KuWiAdv}, that for the nodal structures of $f_{n}$ to exhibit a limit law,\nit is essential to divide $S$ into\nsubsequences whose corresponding $\\rho_{n}$ obey a limiting distribution, i.e. take a subsequence $\\{n\\}\\subseteq S$\nso that $\\rho_{n}\\Rightarrow\\tau$ for some probability measure $\\tau$ on $S^{1}$. In this case\nthe corresponding covariance functions $r_{n}$ converge uniformly locally to $r$, the (inverse) Fourier transform of\n$\\tau$, in the sense of \\eqref{eq:covar func band lim der conv}.\n\n\nLet $\\pazocal{N}(f_{n})$ be the total number of nodal components of $f_{n}$. A combination of the techniques by Nazarov-Sodin\n~\\cite{So12,NS15} and ~\\cite{KuWiAdv} implies that for every sequence $\\{n\\}\\subseteq S$ so that\n\\begin{equation*}\n\\rho_{n}\\Rightarrow \\tau\n\\end{equation*}\nfor some probability measure $\\tau$ on $S^{1}$, there exists a constant $c_{NS}(\\tau)\\ge 0$ so that\n\\begin{equation*}\n{\\mathbf E}[\\pazocal{N}(f_{n})]= c_{NS}(\\tau)\\cdot n +o_{n\\rightarrow\\infty}(1).\n\\end{equation*}\nMoreover, $c_{NS}(\\tau)=0$, if and only if $\\tau$ is either the ``Cilleruelo\" measure\n\\begin{equation*}\n\\tau_{0}=\\frac{1}{4}\\left(\\delta_{\\pm 1}+\\delta_{\\pm i}\\right),\n\\end{equation*}\ni.e. the atomic probability measure supported on the $4$ antipodal points $\\pm 1$ and $\\pm i$ (viewing ${\\mathbb R}^{2}\\cong {\\mathbb C}$),\nor its tilted by $\\pi\/4$ variant\n\\begin{equation*}\n\\widetilde{\\tau_{0}}=\\frac{1}{4}\\left(\\delta_{\\pm \\pi\/4}+\\delta_{3\\pi\/4}\\right).\n\\end{equation*}\nThe probability measures $\\tau_{0}$ and $\\widetilde{\\tau_{0}}$ on $S^{1}$ are the only measures\nsatisfying all the underlying symmetries (invariance w.r.t. rotation by $\\pi\/2$ and complex conjugation) that\nare supported on $4$ points only.\n\n\\vspace{2mm}\n\nThough not applicable as a black box, our Theorem \\ref{thm:main meas conv} (or, rather, the associated techniques)\nyields the following extension of the said results\nconcerning the $V$-direction distribution measures $\\mu_{f_{n}}(V)$ of $f_{n}$\nfor some $V\\in\\mathcal{V}(\\Tb^{2})$ (for example, $V$ could be the constant vector field\n$V\\equiv \\xi$), defined as in \\eqref{eq:mu(f) ddm def}.\nLet $V\\in\\mathcal{V}(\\Tb^{2})$, and $\\{n\\}\\subseteq S$ be a subsequence so that\n\\begin{equation*}\n\\rho_{n}\\Rightarrow \\tau\n\\end{equation*}\nfor some $\\tau\\ne \\tau_{0},\\widetilde{\\tau_{0}}$.\nThen there exists a deterministic probability measure $\\mu_{\\tau;V}$ on ${\\mathbb Z}_{\\ge 0}$, so that\n\\begin{equation*}\n\\mathcal{D}\\left( \\mu_{f_{n}}(V), \\mu_{\\tau;V}  \\right) \\rightarrow 0,\n\\end{equation*}\nas $n\\rightarrow\\infty$. In addition, the support of $\\mu_{\\tau;V}$ is contained in the set of positive even\nintegers $2{\\mathbb Z}_{>0}$.\n\nWe emphasize that for the Arithmetic Random Waves, unlike the band-limited functions in the statement of Theorem \\ref{thm:main meas conv},\nthe limit direction distribution\nmeasure $\\mu_{\\tau;V}$ does depend on $V$, and we are not aware whether it is possible to explicate the dependency of $\\mu_{\\tau;V}$ on\n$V$; this is a by-product\nof the fact that the spectral measure $\\tau$ is not invariant w.r.t. rotations. Moreover, the support might fail to attain the whole of $2\\cdot{\\mathbb Z}_{>0}$,\ndepending on $\\tau$ (and possibly $V$). However, for the generic case \\eqref{eq:rhon equidistr},\nthe conclusions of Theorem \\ref{thm:main meas conv} hold true in their full strength with $\\mu_{d\\theta\/2\\pi}=\\mu_{1}$ same\nas in \\eqref{eq:main meas conv}, and, in particular,\n$\\mu_{d\\theta\/2\\pi}=\\mu_{d\\theta\/2\\pi;V}$ does not depend on $V$, and its support equals to precisely $2{\\mathbb Z}_{>0}$.\n\n\\subsection*{Acknowledgements}\n\nWe warmly thank Ze\\'{e}v Rudnick for suggesting this collaboration, his interest in our work and his support,\nand to Manjunath Krishnapur and Lakshmi Priya for a very enlightening discussion during the conference\n``Random Waves in Oxford\" that took place in June $2018$. Thanks also to Dmitry Beliaev, Matthew de Courcy-Ireland, Gabriel Rivi\\`ere, Nguyen Viet Dang, and Damien Gayet for various discussions on their own work which themselves gave us insights.  A part of the presented research was\nconducted during both authors' visit at the Banff International Research Station (BIRS) in July $2019$,\nand we are grateful to them for providing us with a unique opportunity to use the outstanding facilities of BIRS as part of the\nprogramme ``Research in Pairs\". SE was in residence at McGill University as a Visiting Professor for a majority of the writing of this article and warmly thanks the Department of Mathematics for their hospitality. However, the discussions that lead to the writing of this article began while SE was Lecturer in Analysis at Cardiff University; he is grateful for the opportunity to work in a such stimulating environment. The research leading to these results has received funding from the European Research\nCouncil under the European Union's Seventh Framework Programme (FP7\/2007-2013), ERC grant agreement n$^{\\text{o}}$ 335141 (I.W.).\n\n\n\n\\section{Outline of the proofs and discussion}\n\n\\label{sec:proofs discussions}\n\n\\subsection{Outline of the proofs}\n\n\\label{sec:proofs outline}\n\nIt is known\nthat ~\\cite{Lax,Horm,CH1,CH2, S88, SV96}, under the assumptions above, the covariance functions $K_{T}(\\cdot, \\cdot)$ as in \\eqref{eq:KT covar func}\nscales by $T$ around every point of $\\M$, in the following sense.\nFor $x\\in \\M$, and $u,v\\in {\\mathbb R}^{2}$ so that both $\\frac{\\|u\\|}{T},\\frac{\\|v\\|}{T}$ are less than the injectivity\nradius of $x$ define the {\\em scaled} covariance function using the identification $I_{x}:{\\mathbb R}^{2}\\rightarrow T_{x}\\M$:\n\\begin{equation*}\nK_{T;x}(u,v):= K_{T}(\\exp_{x}(I_{x}(u\/T)),\\exp_{x}(I_{x}(v\/T)))\n\\end{equation*}\non $u,v\\in{\\mathbb R}^{2}$, corresponding to the {\\em scaled} random fields\n\\begin{equation}\n\\label{eq:fxT scaled def}\nf_{x,T}(u)=f_{\\alpha;x,T}(u)=f_{T}\\left(\\exp_{x}(I_{x}(u\/T))\\right),\n\\end{equation}\n$u\\in{\\mathbb R}$ ($\\|u\\|$ smaller than the injectivity radius of $x$).\n\nThen, for all $u,v\\in {\\mathbb R}^{2}$ {\\em fixed},\n\\begin{equation}\n\\label{eq:KTx(u,v)->B(|u-v|)}\nK_{T;x}(u,v)\\rightarrow B(\\|u-v\\|),\n\\end{equation}\nlocally uniformly, where\n\\begin{equation}\n\\label{eq:Bu Fourier annulus}\nB(u):= \\frac{1}{|A_{\\alpha}|}\\int\\limits_{A_{\\alpha}}e^{-2\\pi i \\langle u,v\\rangle} dv\n\\end{equation}\nis the Fourier transform of the characteristic function of the annulus $$A_{\\alpha}=\\{v\\in{\\mathbb R}^{2}:\\: \\alpha<\\|v\\|<1\\}$$\n(the unit circle $S^{1}\\subseteq{\\mathbb R}^{2}$ for $\\alpha=1$),\nand $|A_{\\alpha}|$ is the volume of $A_{\\alpha}$ (resp. the length $\\pi$ of $S^{1}$ for $\\alpha=1$).\nExplicitly, \\eqref{eq:KTx(u,v)->B(|u-v|)} means that for every $R>0$,\n\\begin{equation}\n\\label{eq:covar func band lim der conv}\n\\sup\\limits_{\\|u\\|,\\|v\\|<R} \\left| K_{T;x}(u,v)- B_{\\alpha}(\\|u-v\\|)\\right|\\rightarrow 0,\n\\end{equation}\nand the same holds for all derivatives of $K$, where the rate of convergence in \\eqref{eq:covar func band lim der conv}\ndepends on the order of the derivative only.\n\nThe rotation invariant function $B_{\\alpha}(\\cdot)$ defines a stationary isotropic random field\n$\\mathfrak{g}_{\\alpha}:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$, i.e. the covariance\nfunction of $\\mathfrak{g}_{\\alpha}(\\cdot)$ is $${\\mathbf E}[\\mathfrak{g}_{\\alpha}(x)\\cdot \\mathfrak{g}_{\\alpha}(y)]=B_{\\alpha}(\\|x-y\\|).$$\nBy \\eqref{eq:Bu Fourier annulus}, the spectral measure of $\\mathfrak{g}_{\\alpha}$ is the volume measure of $A_{\\alpha}$ for $\\alpha<1$\n(resp. arc length of $S^{1}$ for $\\alpha=1$).\nIntuitively, \\eqref{eq:covar func band lim der conv} means that there exists a {\\em coupling} of the relevant fields, so that\nthe random fields \\eqref{eq:fxT scaled def} converge, in a sense to be made precise below, to $\\mathfrak{g}_{\\alpha}$.\nNote that the random field $\\mathfrak{g}_{1}$ coincides with Berry's\nRandom Waves Model in \\S\\ref{sec:spher harm RMW}.\n\n\\vspace{2mm}\n\n\nWe follow the general strategy of Nazarov-Sodin ~\\cite{So12,NS15} by first establishing the analogous\nresults for the limit random field $\\mathfrak{g}_{\\alpha}$ (while separating the individual $k$) on ${\\mathbb R}^{2}$,\nthen prove that these extend to $\\M$, locally around every point $x\\in \\M$ (restricted to geodesic balls of\nradius commensurable with $1\/T$), and finally glue all the local results on $\\M$ into a global one.\nCounting tangencies poses a number of marked significant challenges as compared to the original setting\nof Nazarov-Sodin, and also ~\\cite{SW18}.\n\nTo pass from the Euclidean setting into the Riemannian one, a high probability ``stable\" event is created\n(``few\" low lying critical points),\nso that, if it occurs, ``most\" of the nodal components cannot disappear or merge while perturbing the sample function,\nand only few new ones can appear. It was noticed in ~\\cite{SW18} that, on the stable event, not only the number of nodal components is (almost) preserved, but also their topologies. However, the number of $V$-tangencies of a component is {\\em not} a topological\nproperty, and so can vary upon an arbitrarily small perturbation of the sample function. We then have to redefine the stable\nevent to account for possible ``non-transversality\" of the tangencies by translating our problem into one surrounding the quantitative transversal intersection between curves. We also have to use a somewhat\ndifferent analysis for proving that the new stable event is of high probability.\n\nThe reason why the support of $\\mu_{\\alpha}$ does not contain odd integers nor $0$ is that, in the Euclidean setup, if a direction $\\zeta$ is fixed, for a closed simple curve to have $0$ or odd number of $\\zeta$-tangencies would force some of the tangencies to be non-transversal, occurring with\nprobability $0$. Since the Riemannian setup will inherit the direction distribution measures from the Euclidean one, it will also induce the analogous results for the band-limited functions (though having a few nodal components with $0$ or odd number of $V$-tangencies is not dismissed a.s.). The proof for the support being exactly $2 {\\mathbb Z}_+$ follows essentially from a $C^2$ analogue of the barrier method as utilized in \\cite{NS15, SW18, CanSar18} in combination with a necessary $C^2$-closeness estimate for transversal intersections established in \\S\\ref{s:local_results}.\n\nAnother new challenge we encountered is that no analysis of the type explained above is possible around the\n(finitely many) points $x\\in \\M$ where $V(x)=0$.\nTo deal with this situation we will excise small radius-$\\rho$ balls around these problematic points, and bound\nthe contribution of their neighbourhoods to our counts, taking $\\rho\\rightarrow 0$ at the very end. Fortunately,\nit is possible to tune all the parameters encountered and use their relations in our favour. Finally, the local computations with Kac-Rice are useful in order to establish that\nthe limit measure $\\mu_{\\alpha}$ is probability, as their means \\eqref{eq:tot numb tang=sum mean} (or, rather, their Euclidean counterparts in\n\\eqref{eq:sum k*nod(k)<=Ncc} below) stay bounded.\n\n\n\n\n\\subsection{Discussion: further questions}\n\n\n\\subsubsection{Deterministic bounds}\n\nYau's conjecture ~\\cite{Yau1,Yau2} states that if a function $\\phi_{j}:\\M\\rightarrow{\\mathbb R}$ satisfies \\eqref{eq:Helmholts}\non a $d$-manifold $\\M$,\nthen its nodal volume, i.e. the hypersurface volume of the nodal set $\\phi_{j}^{-1}(0)$ is commensurable with $t_{j}$, i.e.\n\\begin{equation*}\nt_{j}\\ll_{\\M}\\operatorname{Vol}(\\phi_{j}^{-1}(0)) \\ll_{\\M} t_{j}.\n\\end{equation*}\nYau's conjecture was settled by Br\\\"{u}ning ~\\cite{Br}, Br\\\"{u}ning-Gromes ~\\cite{BrGr}, and Donnelly-Fefferman ~\\cite{DF}\nfor real analytic manifolds (lower and upper bounds), and, more recently,\na breakthrough progress was made towards the general smooth case ~\\cite{Log Mal,Log Lower, Log Upper}. It would be desirable to\nestablish the analogous deterministic bounds for the direction distribution of the type\n\\begin{equation*}\nt_{j}^{2}\\ll_{\\M} \\mathcal{N}_{V}(\\phi_{j}) \\ll_{\\M} t_{j}^{2}\n\\end{equation*}\nfor a $2$-manifold $\\M$, and $V\\in \\mathcal{V}(\\M)$, in some ``generic\" scenario. Other than\nthe aforementioned bounds for the toral Laplace eigenfunctions \\eqref{eq:fn ARW def}\ndue to Rudnick-Wigman ~\\cite{RudWig2018}, appealing to B\\'ezout's Theorem,\nand the case of spherical harmonics\nwhere one may exploit the fact that these are restrictions of polynomials and can therefore also employ B\\'ezout, this problem is entirely open, at least to our knowledge.\n\n\n\\subsubsection{Uniform convergence w.r.t. $V$}\n\nThe convergence \\eqref{eq:main meas conv} of the direction distribution measure to the limit measure is a priori dependent on $V$.\nHowever, one could explicate the proofs to control the corresponding constants uniformly on compact sets of $\\mathcal{V}(\\M)$ with respect to some reasonable topology that takes into account $V$ and finitely many of its derivatives. One would be interested in characterising the geometry of such vector fields.\n\n\n\\subsubsection{Nonconstant vector fields on ${\\mathbb R}^{2}$}\n\nOur proofs below that $\\mu_{\\alpha}$ is independent of $V\\in\\mathcal{V}(\\M)$ are based\non the fact that if $F:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$ is a stationary isotropic Gaussian random field, then\nthe law of $\\pazocal{N}_{\\xi}(F|_{B(R)})$ (and $\\mu_{F|_{B(R)}}(\\xi)$), corresponding to the restriction of $F$ on a (large)\nball $B(R)$, is independent of the direction $\\xi$. Here $\\xi$ should be\nthought as the direction corresponding to $V(x)$, $x\\in\\M$ is a fixed point on a Riemannian manifold,\nand $F$ should be thought of a scaled version of a random field defined on $\\M$, in the vicinity of $x$.\nThis naturally raises the question whether the same is true for $W\\in\\mathcal{V}({\\mathbb R}^{2})$ in place of $\\xi$,\ni.e. the distribution of $\\pazocal{N}_{W}(F|_{B(R)})$ is independent of $W$, for $W$ that is no longer assumed to be constant,\nat least, asymptotically as $R\\rightarrow\\infty$.\n\n\n\\subsection{Counting version of Theorem \\ref{thm:main meas conv}}\n\nUnlike in Theorem \\ref{thm:main meas conv}, the following Theorem \\ref{t:main_thm_less_equal_1} will separate the counts for different $k$.\nA short argument will then allow to deduce Theorem \\ref{thm:main meas conv} from Theorem \\ref{t:main_thm_less_equal_1}\n(see \\S\\ref{sec:proof main thm meas}).\n\n\\begin{theo} \\label{t:main_thm_less_equal_1}\nLet $V \\in \\mathcal{V}(\\M)$ and $\\alpha \\in [0,1]$. Let $f_{\\alpha, T}$ be the random $\\alpha$-band limited functions \\eqref{eq:fT band lim alpha<1} (or \\eqref{eq:fT band lim alpha=1}) of degree $T$,\nand $\\pazocal{N}_{V}(f_T,k)$ be the number of components whose number of $V$-tangencies is precisely $k$, $k\\ge 0$. Then the following hold:\n\\begin{enumerate}\n\\item There exists $C_{\\alpha, k} \\geq 0$ such that\n\\begin{equation*}\n{\\mathbf E} \\left[ \\left| \\frac{\\pazocal{N}_{V}(f_{\\alpha, T},k)}{\\pazocal{N}(f_{\\alpha, T})} - C_{\\alpha, k} \\right| \\right]\n\\xrightarrow[T \\rightarrow \\infty]{} 0 .\n\\end{equation*}\n\n\\item\nFurthermore, $C_{\\alpha, k} > 0$ for $k \\in 2 {\\mathbb Z}_{> 0}$ and $C_{\\alpha, k} = 0$ otherwise.\n\\end{enumerate}\n\\end{theo}\n\n\n\\section{Direction distribution for Euclidean random fields}\n\n\\subsection{Euclidean random fields: scale invariant model}\n\nThroughout this article we will be interested in centered Gaussian random fields $F: \\mathbb{R}^2 \\rightarrow {\\mathbb R}$.\nBy Kolmogorov's Theorem we know that the law of a centered Gaussian field is determined by its covariance kernel\n\\begin{equation*}\nK(x,y) = {\\mathbf E} \\left[ F(x) \\cdot F(y) \\right].\n\\end{equation*}\nIn particular, we concern ourselves with those fields which are isotropic, that is,\n\\begin{equation*}\nK(x,y) = K(|x-y|),\n\\end{equation*}\nimplying invariance under all rotations and translations. Furthermore, we assume for convenience that $K(0) = 1$.\n\nIt is known that such covariance kernels $K$ can be expressed as the Fourier transform of a measure $\\rho$, called the \\textit{spectral measure} of $F$. In many cases, it is more convenient to describe the field $F$ in terms of $\\rho$ instead of the covariance kernel.  This alternative way of writing $F$ is through the Hilbert space $\\mathcal{H}(\\rho) = \\mathcal{F} \\left( L^2_{sym}(\\rho) \\right)$, that is the space of functions $h$ which are $L^2$-orthonormal with respect to the measure $\\rho$ and satisfy the symmetry rule $\\overline{h}(-Y) = h(Y)$.  In particular, if $\\{\\phi_k\\}_k$ is an orthonormal basis for $\\mathcal{H}(\\rho)$, we can formally write\n\\begin{equation*}\nF = \\sum \\zeta_k \\phi_k\n\\end{equation*}\nwhere $\\zeta_k$  are i.i.d Gaussian random variables. This series diverges a.s. in $\\mathcal{H}(\\rho)$ but under suitable assumptions on $\\rho$, converges a.s. pointwise or in some other sense to a well-defined function, the nature of the convergence depending on the properties of $\\rho$.\n\nFor our purposes, the series of interest converges locally uniformly in $C^k$ for all $k \\in {\\mathbb N}$.\nAs a concrete example, the spectral measure $\\rho$ corresponding to Berry's RWM mentioned in \\S\\ref{sec:spher harm RMW} as the scaling limit\nof $T_{l}$ is the 1-dimensional arc-length measure $ds$ on the circle $S^1$, and we can explicitly compute an orthonormal basis for $\\mathcal{H}(ds)$, so we can express it as\n\\begin{equation*}\n\\sum_{n \\in {\\mathbb Z}} c_n J_{|n|}(r) e^{in \\theta}\n\\end{equation*}\nwhere the $c_k$ are i.i.d. standard complex Gaussians.\n\n\n\\subsection{Kac-Rice method}\n\n\nThe Kac-Rice formula is a standard tool or a meta-theorem for expressing all the moments of {\\em local} quantities, an example being the nodal volume, and the number of critical points of a random field $F:\\mathcal{D}\\rightarrow{\\mathbb R}$ where\n$\\mathcal{D}$ might be a compact subdomain of the Euclidean space or a subdomain of $\\M$. First, for our purposes\nwe need the following two upper bounds on the nodal count, which are direct conclusions from the application of Kac-Rice formula on appropriately defined critical points (either of a function $F$ or its restriction on the boundary of a disc or geodesic ball in $\\M$)\nappearing in ~\\cite{BW17}, which were themselves borrowed from ~\\cite{So12}. These results are valid for all random fields of our interest,\nlying in the $2$ dimensional case only.\n\n\n\\begin{lemma}[Cf. {~\\cite[Lemma 3]{BW17}}, {~\\cite[Corollary 2.3]{So12}}]\n\\label{lem:Kac Rice Euclid}\n\nLet $R>0$ and $F:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$ be a stationary Gaussian random field, so that $F(\\cdot)$ is a.s. $C^{2}$-smooth and so that for every $x\\in {\\mathbb R}^{2}$\nthe vector $\\nabla F(x)$ is non-degenerate Gaussian.\n\n\\begin{enumerate}\n\n\\item Let $\\pazocal{N}(F,R)$ be the number of nodal components of $F$ lying entirely in $B(R)$. Then\n\\begin{equation*}\n{\\mathbf E}[\\pazocal{N}(F,R)] = \\mathcal{O}(R^{2}),\n\\end{equation*}\nwith the constant involved in the $\\mathcal{O}$-notation depending on the law of $F$ only.\n\n\\item Let $\\widetilde{\\pazocal{N}}(F,R)$ be the number of nodal components of $F$ intersecting the circle $\\partial B(R)$. Then\n\\begin{equation*}\n{\\mathbf E}[\\widetilde{\\pazocal{N}}(F,R)] = \\mathcal{O}(R),\n\\end{equation*}\nwith the constant involved in the $\\mathcal{O}$-notation depending on the law of $F$ only.\n\n\n\\end{enumerate}\n\n\n\\end{lemma}\n\nNext, the following result will be useful for ruling out mass leaking for the direction distribution measures (see the proof of\nTheorem \\ref{thm:euclid_count} (4) in \\S\\ref{sec:proof thm euclid no leak} below). It is a direct consequence of ~\\cite[Theorem 6.3]{AW}.\n\n\\begin{lemma}\n\\label{lem:Kac-Rice Euclid tangencies}\nLet $F:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$ be a stationary Gaussian random field, so that the random vector $(F(0),\\nabla F(0))$\nhas a non-degenerate $3$-variate Gaussian distribution, $\\zeta\\in S^{1}$,\nand denote $\\mathcal{N}_{\\zeta}(F,R)$ to be the number of $\\zeta$-tangencies of $F$. Then\n\\begin{equation*}\n{\\mathbf E}[\\mathcal{N}_{\\zeta}(F,R)] = C_{0}\\cdot \\operatorname{Vol}(B(R)),\n\\end{equation*}\nfor some $C_{0}>0$ depending on the law of $F$ only.\n\\end{lemma}\n\nFinally, the following lemma yields an upper bound for the number of nodal components lying in a small geodesic ball\nfor the band limited functions only.\n\n\n\\begin{lemma}[{~\\cite[Lemma 2]{So12}}]\n\\label{lem:Kac Rice manifold}\nLet $\\alpha \\in [0,1]$ and $f_{T}=f_{\\alpha;T}$ be the random band-limited function \\eqref{eq:fT band lim alpha<1}\n(or \\eqref{eq:fT band lim alpha=1}).\nFor $x\\in \\M$ and $r>0$ sufficiently small, denote $\\pazocal{N} (f_{x,T},r)$ to be the number of nodal components of $f_{T}$ lying inside\nthe geodesic ball $B(x,r)\\subseteq\\M$ centred at $x$ with radius $r$. Then their expected number satisfies the\nfollowing estimate:\n\\begin{equation*}\n{\\mathbf E}[\\pazocal{N} (f_{x,T},r)] = \\mathcal{O}(T^{2}\\cdot r^{2}),\n\\end{equation*}\nwith constant involved in the $\\mathcal{O}$-notation depending on $\\M$ and $\\alpha$ only.\n\\end{lemma}\n\n\n\\subsection{Direction distribution for Euclidean random fields}\n\nLet $F:{\\mathbb R}^{2}\\rightarrow{\\mathbb R}$ be a smooth Gaussian random field with spectral measure $\\rho$,\n$\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ be non-vanishing, and $k \\in {\\mathbb Z}_{\\geq 0}$. We denote by\n\\begin{equation*}\n\\pazocal{N}_{\\tilde{V}}(F,k,R)\n\\end{equation*}\n(resp. $\\pazocal{N}(F,R)$) the number of connected components of $F^{-1}(0)$ (which are all smooth thanks to Bulinskaya's Lemma \\cite[Proposition 6.11]{AW}) that are both completely contained in $B(R)$ and whose number of $\\tilde{V}$-tangencies is precisely $k$\n(resp. the total number of connected components of $F^{-1}$ contained in $B(R)$).\n\n\nAs we would like to study the distribution of $\\pazocal{N}_{\\tilde{V}}(F, \\cdot, \\cdot)$, it is essential to show that this function is in fact a random variable (i.e. that it is measurable on the given sample space of Gaussian fields $F$). A detailed verification of this fact for the related random variable $N(F, R, \\mathcal{T})$, that is the number of connected components contained in the ball $B(R)$ with topological type $\\mathcal{T}$, was carried out in the Appendix of \\cite{SW18} and is sufficiently robust to prove measurability in our case. We leave the details to the interested reader. We now state some axioms on $F$ (rather, its spectral measure $\\rho$) so that to be able to formulate the main result (Theorem \\ref{thm:euclid_count} immediately below)\non Euclidean random field.\n\n\\begin{definition}[Axioms on $\\rho$]\n\\label{def:axiom spec meas}\nFor a given Gaussian stationary field $F$, we will sometimes impose the following restrictions on the corresponding spectral measure $\\rho=\\rho_{F}$:\n\\begin{itemize}\n\\item $(\\rho 1)$ The measure $\\rho$ has no atoms (if and only if the action of the translations is ergodic by Theorem \\ref{thm:GFM}).\n\\item $(\\rho 2)$ For some $p > 4$,\n\\begin{equation*}\n\\int\\limits_{{\\mathbb R}^2} |\\lambda|^{p} \\, d \\rho(\\lambda) < \\infty\n\\end{equation*}\n(this ensures that a.s. $C^2$-smoothness of $F$).\n\\item $(\\rho 3)$ The support of $\\rho$ does not lie in a linear hyperplane (this ensures that the Gaussian field, together with its gradient, is not degenerate).\n\n\\item $(\\rho 4^*)$ The support of $\\rho$ has non-empty interior.\n\n\\end{itemize}\n\\end{definition}\n\n\nNazarov and Sodin ~\\cite{So12,NS15}\nproved that if $F$ satisfies the axioms $(\\rho 1)-(\\rho 3)$, then there exists a constant $c(\\rho)=c_{NS}(\\rho)\\ge 0$\n(``Nazarov-Sodin constant \"), so that\n\\begin{equation}\n\\label{eq:NS const def prop}\n{\\mathbf E}\\left[\\left|\\frac{\\pazocal{N}(F,R)}{\\operatorname{Vol}(B(R))} - c(\\rho) \\right|\\right] \\rightarrow 0.\n\\end{equation}\nIf, further, $(\\rho 4^{*})$ is satisfied, then $c(\\rho)>0$. In fact, in most of what follows, we will work with the following weaker version\nof $(\\rho 4^{*})$.\n\n\n\\begin{definition}[Axiom $(\\rho 4)$]\n\n$(\\rho 4)$ The Nazarov-Sodin constant $c(\\rho)$ is positive.\n\n\\end{definition}\n\n\n\n\\begin{theo} \\label{thm:euclid_count}\nLet $F: {\\mathbb R}^n \\rightarrow {\\mathbb R}$ be a stationary random field whose spectral measure $\\rho$ satisfies axioms $(\\rho 1) - (\\rho 4)$,\n$k \\ge 0$, $\\zeta\\in{\\mathbb R}^{2}$ be a fixed direction, and recall the Nazarov-Sodin constant $c(\\rho)>0$ satisfying the defining\nproperty \\eqref{eq:NS const def prop}. Then:\n\\begin{enumerate}\n\\item There exists $C_{k, \\zeta}=C_{k, \\zeta}(\\rho) \\geq 0$\n\\begin{equation}\n\\label{eq:conv L1 Euclid}\n{\\mathbf E} \\left[ \\left| \\frac{\\pazocal{N}_{\\zeta}(F,k,R)}{c(\\rho)\\cdot \\operatorname{Vol} (B(R))} - C_{k, \\zeta} \\right| \\right] \\rightarrow 0\n\\end{equation}\nas $R \\rightarrow \\infty$. If $k=0$ or $k$ is odd, then $C_{k,\\zeta}=0$.\n\n\\item Assuming further that $F$ is isotropic, we have that $C_k=C_{k, \\zeta}$ is independent of $\\zeta$.\n\n\\item If $k\\in 2{\\mathbb Z}_{>0}$, and either $F$ satisfies the axiom $(\\rho 4^*)$ (without assuming that $F$ is isotropic),\nor $\\rho = \\sigma_{S^1}$ (i.e. $F$ is Berry's monochromatic isotropic random waves),\nthe normalized arc-length measure on $S^1$, then $C_{k,\\zeta} > 0$.\n\n\\item We have\n\\begin{equation*}\n\\sum\\limits_{k=0}^{\\infty}C_{k,\\zeta} = 1.\n\\end{equation*}\n\n\\end{enumerate}\n\\end{theo}\n\n\\section{Various facts from plane geometry}\n\n\\subsection{Vector fields and exponential maps}\n\n\\begin{defn} \\label{d:blown_up_V}\nLet $F  \\in C^{\\infty}(T_x\\M)$ and $T>0$. For $V \\in \\mathcal{V}(\\M)$ and $Y \\in T_x\\M$, we define the \\textit{blown-up (at scale $T$) vector field} $\\tilde{V}_{x,T} \\in \\mathcal{V}(T_x\\M)$ at $x \\in \\M$, where $exp_x(0) = x$ and $exp_x(Y)=y$, via the local coordinate formula\n\\begin{equation}\n\\label{eq:tild(V) def}\n(\\tilde{V}_{x,T} F) =\n a_1 \\left( \\exp_x \\left( \\frac{Y_1}{T}, \\frac{Y_2}{T} \\right) \\right) \\cdot \\frac{\\partial F}{\\partial Y_1}\\left( Y  \\right)\n+ a_2 \\left( \\exp_x \\left( \\frac{Y_1}{T}, \\frac{Y_2}{T} \\right) \\right) \\cdot \\frac{\\partial F}{\\partial Y_2} \\left( Y \\right).\n\\end{equation}\n\\end{defn}\n\nWe will shortly consider the random fields\n$F(Y_1, Y_2) = f_{\\alpha, T}(\\exp_x(\\frac{Y}{T}))$ as in \\eqref{eq:fxT scaled def}. Notice that we are pushing forward the vector field $V$, extracting the ``top order in $T^{-1}$\" differential operator, and then applying that to $F$. We {\\em are not} pushing forward $V$ by $M_{1\/T} \\circ (\\exp_x)^{-1}_*$, where $M_{1\/T}$ is simply multiplication in the fibers of $T_xM$ by $1\/T$. The reason for this particular definition is so that the coefficients of $V$ are localizing at the same scale as $f_{x, T}$.\nThus in the regime $|Y| \\leq R$ with $R\/T = o(1)$, for a large but fixed $R$ with $T \\rightarrow \\infty$, we find that the coefficients $a_1,a_2$ become constants up to first order in $T^{-1}$. More specifically, we have that, for instance, $a_1 \\left( \\exp_x(\\frac{Y_1}{T}, \\frac{Y_2}{T}) \\right) = a_1(x) + \\mathcal{O} \\left( \\frac{|Y|}{T} \\right)$ with $\\frac{|Y|}{T} = o(1)$ as seen in upcoming sections discussing limiting regimes.\n\nFrom here onward, we reserve the notation of $\\tilde{V}$ for vector fields on ${\\mathbb R}^2$. Now, in order to show that our main result Theorem \\ref{t:main_thm_less_equal_1} is not vacuous, we give the following result on the existence of vector fields with finitely many zeroes.\n\n\\begin{lem} \\label{l:vfs_finite_zeroes}\nGiven a smooth and compact surface $(\\M,g)$, there exists smooth vector fields $V$ that have finitely many zeroes. That is, our class $\\mathcal{V}(\\M) \\neq \\emptyset$ for compact Riemannian surfaces $\\M$.\n\\end{lem}\n\nThis result holds in $d\\ge 2$ dimensions, but we state it for surfaces given our context.\n\n\\begin{proof}\nConsider any Morse function $f$ on $\\M$; we know there exists such a function, for example one being the height function on $\\M$ after applying an appropriate embedding into ${\\mathbb R}^N$. The metric $g$ allows us to associate the differential $df$ to its gradient field $\\nabla_g f$, each of whose zeroes is isolated by the non-degenerate critical point condition. Compactness of $\\M$ yields that the number of such zeroes must be finite.\n\\end{proof}\n\n\\subsection{Global plane geometry}\n\nIn this section we begin our study of curves in ${\\mathbb R}^2$ that have a fixed number of tangencies to a given vector field $\\tilde{V}$ and its relation to the study of transversality between the sets $f^{-1}(0)$ and $(Vf)^{-1}(0)$. These elementary results, although simple to state and are intuitively clear, prove very useful in the pursuit of Theorem \\ref{t:main_thm_less_equal_1}. We begin with a simple but crucial lemma that will play a key role in showing that the limiting constants $C_k$, for $k=0,1$, in Theorem \\ref{t:main_thm_less_equal_1} are $0$:\n\n\\begin{lem} \\label{l:at_least_two_tangencies}\nSuppose $\\zeta$ is a fixed direction on ${\\mathbb R}^2$. Then any closed simple curve in the plane must have at least two tangencies $\\zeta$.\n\\end{lem}\n\nBefore proving this, we remind ourselves of a simple form of the Gauss-Bonnet Index Theorem, which goes by the name of the Theorem of Turning Tangents in elementary differential geometry, due to Hopf:\n\n\\begin{theo}[Theorem of Turning Tangents] \\label{t:turn_tang}\nThe rotation number of an embedded $S^1$ in ${\\mathbb R}^2$ (i.e. the oriented number of times the unit tangent vector $\\overrightarrow{T}$ to this embedding transverses $S^1$) is equal to $\\pm 1$.\n\nIn other words, the map induced by $\\overrightarrow{T}$ from $S^1$ to $S^1$ must be surjective and in terms of topological language, $\\overrightarrow{T}$ induces a map of degree $\\pm 1$.\n\\end{theo}\n\n\\begin{proof}[Proof of Lemma \\ref{l:at_least_two_tangencies}]\nBy Theorem \\ref{t:turn_tang}, $\\overrightarrow{T}$ must be parallel to $\\zeta$ and -$\\zeta$ at at least two distinct points.  Since $\\overrightarrow{T}$ parallel to -$\\zeta$ is counted as a $\\zeta$-tangency, this proves our claim.\n\\end{proof}\n\nIt is possible to construct closed curves in ${\\mathbb R}^2$ with an odd number of tangencies:\nall we must do is properly glue together neighbourhoods of the singular parts of a cubic to the neighbourhoods of singular parts of the quadratics. However, we will see that this situation is unstable in an appropriate sense, and therefore will not contribute towards our final result.\n\n\\begin{lem} \\label{l:even_int}\nConsider two $C^2$ curves $\\gamma_1, \\gamma_2 \\subsetneq {\\mathbb R}^2$ that arise as components of the zero sets of respectively two different functions, both of which has 0 as a regular value. Assume that $\\# ( \\gamma_1 \\cap \\gamma_2)  < \\infty$.  If $\\gamma_1$ is both simple and closed or $\\gamma_2$ is both simple and closed, and each point in $\\gamma_1 \\cap \\gamma_2$ is transversal, then $\\# \\gamma_1 \\cap \\gamma_2$ is even.\n\\end{lem}\n\n\\begin{proof}\nWithout loss of generality, let $\\gamma_1$ be simple and closed.  By the Jordan Curve Theorem, $\\gamma_1$ encloses a bounded region $\\Omega_1$.  By the transversality assumption, every point of intersection is uniquely connected to another intersection  by an arc  of $\\gamma_2$ that is completely contained in $\\Omega_1$ (for otherwise, this contradicts the regularity assumption or the transversality assumption).  Hence, we can associate two unique points of intersection to each interior arc.  Counting the number of such arcs, we arrive at our conclusion.\n\\end{proof}\n\n\\begin{cor}\n\\label{cor:tangencies even}\nFor a fixed direction $\\zeta$ on ${\\mathbb R}^2$, any simple closed curve in the plane without inflection points that are also $\\zeta$-tangent must have an even positive number of $\\zeta$-tangencies.\n\\end{cor}\n\nWe close this section by emphasizing that by the qualitative version of Bulinskaya's Lemma (see \\cite[Proposition 6.11]{AW}),\nwith probability 1 any nodal component of a Gaussian field $F$ on $B(0,R)$ is simple and regular (that is 0 is regular value for $F$) although there may exists those that are not closed.\n\n\\subsection{Intersections in the plane: local results} \\label{s:local_results}\n\nThe terminology established in the section is important from both analytic and geometric points of view. It is here where we establish our main observation: \\textit{the study of nodal curves with a fixed number of $V$-tangencies is equivalent to the study of the transversal and tangential intersections of the sets $f_{T, \\alpha}^{-1}(0)$ and $(Vf_{T, \\alpha})^{-1}(0)$.}\n\nOur aim is to interchange the words of ``intersection\" and ``$V$-tangency\" with the latter being a special case of the former in the upcoming dictionary. Although this may appear counter-intuitive at first, it is crucial that we identify our geometric property with a suitable analytic characterization.\n\n\\subsubsection{Deterministic intersection results: $\\beta$-transversal tangencies} \\label{s:deterministic}\n\n\\begin{defn} \\label{d:beta_trans}\n\\begin{enumerate}\n\\item For a pair of functions $G=(g_1, g_2): {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ such that $g_1, g_2 \\in C^1(B(2R))$, $0$\nis a regular value for both $g_1$ and\n$g_2$, and $\\beta>0$ (possibly small), we say that $u_0$ is a point of \\textit{$\\beta$-transverse intersection} if $g_1(u_0)=g_2(u_0) = 0$ and $\\left| \\det DG(u_0) \\right| >  \\beta$.\n\n\\item Let $\\beta>0$ be a small parameter.\nGiven $g \\in C^2({\\mathbb R}^2)$ such that $0$ is a regular value with $\\tilde{V}$ a smooth vector field on ${\\mathbb R}^2$, we say that $u_0$ is a \\textit{$\\beta$-transverse $\\tilde{V}$-tangency} (or simply a ``non-degenerate\" tangency) if $\n\\tilde{V}(u_0) \\neq 0 \\in {\\mathbb R}^2$, $g(u_0)=\\tilde{V}g(u_0)=0$, and $\\left| \\det \\left( D_{u=u_0} \\left[ \\nabla g, \\nabla \\tilde{V}g \\right] \\right) \\right| > \\beta$.\n\\end{enumerate}\n\\end{defn}\n\nNow that we have fixed some useful terminology, we proceed to establishing some key yet general estimates on the intersections of curves in the planes and their behaviors under perturbations.  In this section, we analyse a situation of quantitative transversality:\n\n\\begin{prop}[Deterministic stability of transversal intersections, cf. {\\cite[Lemma 10]{BW17}}] \\label{p:stab_trans}\nLet $R>0$ and $k \\ge 0$ be given. Let $\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ be non-vanishing, $g,h \\in C^3(\\overline{B}(2R))$, and denote $G:=(g, \\tilde{V}g)$. Assume that the following restrictions hold:\n\\begin{enumerate}\n\\item There exists $\\beta_1 >0$ such that\n\\begin{equation*}\n\\min_{\\overline{B}(2R)} \\max \\{ |g(u)|, |\\nabla g(u)| \\}, \\,  \\min_{\\overline{B}(2R)} \\max \\{ |\\tilde{V}g(u)|, |\\nabla (\\tilde{V} g)(u)| \\} > \\beta_1.\n\\end{equation*}\nThis immediately implies that\n\\begin{equation*}\n\\min_{\\overline{B}(2R)} \\max \\{ \\|G\\|, \\|DG\\|_{\\mathcal{L}({\\mathbb R}^2 \\rightarrow {\\mathbb R}^2)} \\} > \\frac{\\beta_1}{\\sqrt{2}},\n\\end{equation*}\nand in particular that $\\|g\\|_{C^2} > c_2\\beta_1$ where $c_2>0$ is a constant depending only on the ambient dimension.\n\\item There exists $\\beta_2$ such that\n\\begin{equation*}\n\\left| \\det DG(u) \\right| >  \\beta_2\n\\end{equation*}\nfor $u \\in \\{ (g, \\tilde{V}g) = 0 \\}$ in $B(2R)$.\n\\item The $C^3$ norms of $g,h$ are bounded above by $M_0$ for some $M_{0}>0$.\n\\item We have\n\\begin{equation*}\n\\| g - h \\|_{C^2(B(R+1))} < b_0\n\\end{equation*}\nfor some $b_{0}=b_{0}(M_{0},\\beta_{0},\\beta_{1})>0$ sufficiently small.\n\n\\end{enumerate}\n\nThen there exists an injective map $\\gamma \\rightarrow \\gamma^h$ between the components of $g^{-1}(0)$ and $h^{-1}(0)$ with the property that:\n\\begin{enumerate}\n\\item The connected components $\\gamma \\subset B(R-1)$ of $g^{-1}(0)$ whose number of $\\beta$-transversal $\\tilde{V}$-tangencies are precisely\n$k$ is in one-to-one correspondence with a subset of the connected components $\\gamma^{h} \\subset B(R)$ of $h^{-1}(0)$ whose number of $(\\beta-b)$-transversal $\\tilde{V}$-tangencies is precisely $k$.\n\n\\item For every $\\gamma$ as above, the components $\\gamma$ and $\\gamma^h$ are uniformly close in the following sense: there exists a smooth bijective map $\\psi_{\\gamma}: \\gamma \\rightarrow \\gamma^h$ such that for all $u \\in \\gamma$, we have the estimate\n\\begin{equation} \\label{e:components_bouge}\n\\| \\psi_{\\gamma}(u) - u \\|_{L^{\\infty}(B(R))} = \\mathcal{O}(b)\n\\end{equation}\nwhere the implicit constant depends on the quantities $\\|G\\|_{C^2(B(0,2R))}$ and $\\beta_1$.\n\\end{enumerate}\n\\end{prop}\n\nThe dependence of the implicit constant in the estimate \\eqref{e:components_bouge} for the new zero's positioning \\textit{linear} in $\\beta_1$ but of order $-1$ in $\\beta_2$.\n\n\n\\subsubsection{Some technical lemmas}\n\nIn this section we give a key lemma following from multivariable calculus, which is essentially a restatement of the above proposition, and is analogous to \\cite[Lemma 7]{So12}.\n\n\\begin{lem} \\label{l:Sodin_type_calc_lem}\nConsider $G(u) = (g_1(u), g_2(u))$ for $g_1, g_2 \\in C^2(B(2R)$ for some given $R>0$ possibly large. Suppose $u_0 \\in G^{-1}(0) \\cap B(2R)$. We make the following assumptions:\n\\begin{itemize}\n\\item There exists $1 > a>0$ such that for all $u \\in B(2R)$, we have\n\\begin{equation} \\label{e:no_low_lying_cp}\n\\min_{\\overline{B}(2R)} \\max \\{ |g_1(u)|, |\\nabla g_1(u)| \\}, \\,  \\min_{\\overline{B}(2R)} \\max \\{ |g_2(u)|, |g_2(u)| \\} > a.\n\\end{equation}\n\\item There exists $\\alpha> 0$ such that $\\alpha < |\\det DG(u)|$ for all $u \\in B(2 R) \\cap G^{-1}(0)$.\n\\item Set\n\\begin{equation} \\label{e:perturb_size}\n b =\\frac{1}{4} \\min \\left\\{ a , \\frac{1}{ \\|G\\|_{C^2(B(u_0, R_0+1))}}, \\alpha \\right\\}.\n\\end{equation}\n\n\\end{itemize}\nLet $\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ be non-vanishing.  Consider a perturbation $\\Psi = (\\psi, \\tilde{V} \\psi)$ where $\\psi \\in C^2(B(2R))$.  Set $G_{\\Psi}(u) = G(u) + \\Psi(u)$ and consider only zeroes $u_0 \\in G^{-1}(0)$ such that $B(u_0, b) \\subset B(0, R_0)$.  Then for $\\| \\psi \\|_{C^2(B(2R))} < b$ we have that\n\\begin{enumerate}\n\\item $u_0$ is a isolated zero of $G$,\n\\item $G_{\\Psi}$ is a global diffeomorphism on $B ( u_0,  b )$,\n\\item $G_{\\Psi}$ also has a unique isolated zero $\\tilde{u_0} \\in B(u_0, b) $ with\n\\begin{equation*} \\label{e:new_zero_pos}\n\\|u_0 - \\tilde{u_0} \\| =  \\mathcal{O}(b)\n\\end{equation*}\nwhere the implicit constant depends on the quantities $\\|G\\|_{C^2}$ and $a^{-1}$.\nMoreover, the new zero $\\tilde{u_0}$ has the property that $|det DG_{\\psi}(\\tilde{u_0})| > \\alpha - b.$\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nEach of these claims follows almost directly from the Inverse Function theorem and its proof. Part (1) is an immediate implication of the inverse function theorem.\n\nParts (2) and (3) use elements from the proof of the Inverse Function theorem. Here, we closely follow \\cite[Theorem 22.26]{B03}. If we can show that the so-called \\textit{Newton's function} $\\epsilon_{\\Psi}(u) = DG_{\\Psi}^{-1}(u_0)G_{\\Psi}(u+u_0) - u$ has the property $\\|D\\epsilon_{\\Psi}(u)\\|  < 1$ for all $u \\in B(0, b)$ with $b$ satisfying the stated inequality, the referenced proof of the Inverse Function theorem shows that $\\epsilon_{\\Psi}$ is invertible on $B(0, b)$ and therefore $G_{\\Psi}$ is invertible on $B(u_0, b)$. Thus, we estimate the operator norm for the matrix $D \\epsilon_{\\Psi}$:\n\\begin{align*}\n& \\left( DG(u_0) + D\\Psi(u_0) \\right)^{-1} \\left( DG(u+u_0) + D\\Psi(u+u_0)  \\right) - I \\\\\n& = \\left( DG(u_0) + D\\Psi(u_0) \\right)^{-1} \\left( DG(u_0) + D\\Psi(u+u_0) + \\mathcal{O}_{\\|G\\|_{C^2(B(2R))}}(\\|u\\|)  \\right) - I \\\\\n&  = \\left(I + DG(u_0)^{-1} D\\Psi(u_0) \\right)^{-1} (DG(u_0))^{-1} \\left( DG(u_0) + D\\Psi(u+u_0) +  \\mathcal{O}_{\\|G\\|_{C^2(B(2R))}}(\\|u\\|) \\right) - I  \\\\\n& = \\left( I + DG(u_0)^{-1} D\\Psi(u_0) \\right)^{-1} \\left( I + DG(u_0)^{-1} D\\Psi(u+u_0) + \\mathcal{O}_{ \\|G\\|_{C^2(B(2R))}}(a^{-1} b )  \\right) - I \\\\\n&= \\mathcal{O}\\left(a^{-1} \\cdot b \\right) +  \\mathcal{O}\\left( \\|G\\|_{C^2(B(u_0, b))} \\cdot a^{-1} \\cdot b \\right) +  \\mathcal{O} \\left((a^{-1}b)^2 \\right) \\\\\n& = \\mathcal{O}_{a, \\|G\\|_{C^2}}(b),\n\\end{align*}\nprovided we have the restriction \\eqref{e:perturb_size} and $a < 1$, where we have used some well-known formulas for inverses of small perturbations of the identity matrix. This in turn implies that the inverse function $G_{\\psi}^{-1}$ exists on $B(u_0, b)$, therefore establishing claim (2).  Hence any zero, if it exists, lying inside of $B(u_0, b)$ must be unique and isolated.  We have now reduced ourselves to proving the existence of a new zero $\\widetilde{u_0}$.\n\nThe proof of the existence and exact positioning of the isolated zero is motivated by the proof of \\cite[Lemma 10]{BW17}.\nAs $N_i(u_0) = \\frac{\\nabla g_i(u_0)}{\\| \\nabla g_i(u_0)\\|}$ is well-defined thanks to $|\\det DG(u_0) | > \\alpha$, we consider the function\n\\begin{equation*}\n\\zeta(s_1, s_2) := G_{\\psi}\\left(u_0 + s_1 N_1(u_0) + s_2 N_2(u_0) \\right);\n\\end{equation*}\nwe would like to show that $\\zeta$ does obtain a zero $\\tilde{s}_0$ and give an estimate on $\\|\\tilde{s}_0\\|$. This would establish claim (3).\nTo begin, consider the auxiliary function\n\\begin{equation*}\n\\mathfrak{S}_{(t_1, t_2)}(s_1, s_2) := (t_1, t_2) - \\epsilon_{\\zeta}(s_1,s_2)\n\\end{equation*}\nwhere $\\epsilon_{\\zeta}(s_1, s_2) = D\\zeta^{-1}(0,0) \\zeta(s_1,s_2) - (s_1, s_2)$ and $(t_1,t_2)$ are small numbers; we will employ an effective version of the Contraction Mapping Theorem in order to determine the existence and size of $\\tilde{s}_0$.\n\nLet us take a pause to understand the purposes of our various newly-defined auxiliary functions.  Supposing that $\\epsilon_{\\zeta}(s_1, s_2)$ were well-defined and that we were to able solve the fixed point equation\n\\begin{equation} \\label{e:fixed_point_H}\n\\mathfrak{S}_{(t_1,t_2)} (H(t_1,t_2)) = H(t_1,t_2)\n\\end{equation}\nfor $(t_1, t_2)$ varying over some small neighborhood around the origin. This would imply that $H(0,0) = - \\epsilon_{\\zeta}(H(0,0))$. As $D\\zeta^{-1}(0,0)$ is invertible our assumption $\\epsilon_{\\zeta}$ being well-defined, then this along with the equation $H(0,0) = - \\epsilon_{\\zeta}(H(0,0))$ would show that $ D\\zeta^{-1}(0,0) \\zeta(H(0,0))=(0,0)$.  Thus $ \\zeta(H(0,0))=(0,0)$ and moreover $\\zeta$ obtains a zero.  If we can show that  $$\\|D_{s_1,s_2} \\mathfrak{S}_{(t_1,t_2)}(s_1, s_2) \\| < 1$$ for suitably small values of $(t_1,t_2)$ and $(s_1,s_2)$, then the Contraction Mapping Theorem as used in the proof of the Inverse Function Theorem gives us such an $H(t_1,t_2)$ as in (\\ref{e:fixed_point_H}) and therefore $\\tilde{s}_0:=H(0,0)$ is achieved.\n\nTo sum up our reasoning, first we must estimate the operator norm of $D_{s_1, s_2}\\mathfrak{S}_{(t_1, t_2)}$ for $\\|(t_1,t_2)\\|, \\|(s_1,s_2)\\| \\leq Cb$ to show the existence of $H(s_1,s_2)$ which in turn gives the existence of a zero $\\tilde{s}_0 = H(0,0)$. Second, an estimate on $\\|\\epsilon_{\\zeta}(s_1,s_2)\\|$ for $\\|(s_1, s_2)\\| \\leq C b$ where $C>0$ is some uniform constant gives us the positioning of $\\tilde{s}_0$; this follows from the equation $H(0,0) = - \\epsilon_{\\zeta}(H(0,0))$. We now proceed to carrying out these steps.\n\nA direct calculation will show that\n\\begin{equation*}\nD_{s_1,s_2} \\zeta_{|(0,0)} = D_{s_1,s_2} G_{\\Psi} \\left( u_0 + s_1 N_1(u_0) + s_2 N_2(u_0)\\right)_{|(0,0)}\n\\end{equation*}\nis invertible. Thanks to $G$ being a local diffeomorphism in $B(u_0, b)$ (and therefore $\\{N_1, N_2\\}$ forming a linearly independent set over $B(u_0, b)$), this combined with  $|\\det (DG + D\\psi)(u)| \\geq \\alpha - 2b > 0$, which follows by (\\ref{e:no_low_lying_cp}) and (\\ref{e:perturb_size}), for all $u \\in B(u_0, b)$ finally shows that $D\\zeta(0,0)$ is invertible. Thus, we have an expression\n\\begin{align*}\n&  D_{s_1,s_2} \\mathfrak{S}_{(t_1,t_2)}(s_1, s_2) = -D \\epsilon_{\\zeta}(s_1,s_2) \\\\\n& = \\left[N_1(u_0), N_2(u_0) \\right]^{-1} \\circ DG^{-1}_{\\psi}(u_0) \\circ DG_{\\psi}(u_0 + sN_1 + sN_2) \\circ \\left[N_1(u_0), N_2(u_0) \\right] - I\n\\end{align*}\nwhere we arrive at a similar Taylor expansion analysis as carried out in the first half of our proof. We leave it to the interested reader to see that for $\\|D_{s_1,s_2} \\mathfrak{S}_{(s_1,s_2)}(s_1, s_2)\\| = \\mathcal{O}(b) < 1$, where the implicit constant again depends on $\\|G\\|_{C^2}$ and $a^{-1}$.  Another application of the Contraction Mapping Theorem, thanks to the operator norm of $D_{s_1,s_2} \\mathfrak{S}_{(t_1,t_2)}(s_1, s_2)$ being smaller than 1, brings us to the conclusion that such a solution $H$ exists.\n\nWe know that $\\epsilon_{\\zeta}(s_1, s_2) = \\epsilon_{\\zeta}(0, 0) + \\mathcal{O}_{\\|D\\epsilon_{\\zeta}(0,0)\\|}(b)$. Furthermore,\n$\\epsilon_{\\zeta}(0,0) = D \\zeta^{-1}(0,0) \\Psi(u_0)$ since $u_0$ is a zero of $G$. Finally as $\\|\\psi\\|_{C^2} < b$, we conclude that $\\tilde{s}_0 = H(0,0) =  \\mathcal{O}(b)$ with the corresponding implicit constants.\n\\end{proof}\n\nNote that our application does require $C^2$ regularity due to setting $g_2 = \\tilde{V}_{x,T}g_1$. Our proof demonstrates that the amount in which our zeroes move after perturbation is controlled by the $C^1$ norm of the perturbation $\\Psi$ (thanks to our perturbation being simply additive in nature), emphasizing that from the point of view of analysing zeroes the $C^1$ approximation of the scaling limit $\\mathfrak{g}_{\\alpha}$ is sufficient.\n\n\\begin{proof}[Proof of Proposition \\ref{p:stab_trans}]\nSet $g = g_1$ and $\\tilde{V}g = g_2$. This demonstrates why $C^2$ estimates are imposed. Now, we simply set $\\beta_1 = a$, $\\beta_2 = \\alpha$, and take $b$ equal to the strength of the perturbation. The final step involves applying in tandem \\cite[Lemma 10]{BW17} (although we can easily repeat the steps of the proof of Lemma \\ref{l:Sodin_type_calc_lem} to recover this result) and conclusion (3) of Lemma \\ref{l:Sodin_type_calc_lem} to show that non-degenerate components map to a subset of non-degenerate components and stay $\\mathcal{O}(b)$ close in $L^{\\infty}$.\n\\end{proof}\n\nA combination the Proposition \\ref{p:stab_trans} and \\cite[Lemma 7]{So12} immediately gives us that:\n\n\\begin{cor} \\label{l:Sodin_annuli}\nUnder the hypotheses of Lemma \\ref{l:Sodin_type_calc_lem}, let $C>0$ be the implicit constant that appears in the estimate (\\ref{e:new_zero_pos}).  Take $\\Gamma_g$ a regular component of $g^{-1}(0)$. Then, if given a tubular neighborhood $A_{\\Gamma_g}(Cb)$ that uniquely contains the component $\\Gamma_g \\subset g^{-1}(0)$ and has width $2Cb$, it follows that the points of $\\{ (g, \\tilde{V}g) = 0 \\}$ which meet $\\Gamma_g$ persist inside of $A_{\\Gamma_g}(Cb)$ after perturbation by $h-g$.\n\\end{cor}\n\n\n\\subsubsection{Deterministic intersection results: sub-$\\beta$ tangencies}\n\nThe main purpose of this section is to provide the reader with the intuition that components with ``near-tangential\" intersections can be unstable in terms of intersection counts being constant when subjected to small perturbations. The following lemma will not be employed in the proofs of our main results, however, it serves as an intuition what might happen to the tangency points if we do not exclude the unstable event $\\Delta_7$, as\ndefined in \\S\\ref{s:exceptional_events} below.\n\n\\begin{defn} \\label{d:sub_beta}\n\\begin{enumerate}\n\\item For a function $G=(g_1, g_2)$ such that $g_1, g_2 \\in C^1(B(2R))$ and $\\beta>0$ (possibly small), we say that $u_0$ is a point of \\textit{sub-$\\beta$ intersection} if $G(u_0) =  (g_1(u_0), g_2(u_0)) = 0$ and $\\left| \\det DG(u_0) \\right| \\leq  \\beta$.\n\n\\item Let $\\beta>0$ be a small parameter. Given $g \\in C^2({\\mathbb R}^2)$ with $\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ non-vanishing, we say that $u_0$ is a point of \\textit{sub-$\\beta$ $\\tilde{V}$-tangency} (or simply a ``near-degenerate\" tangency) if $( g(u_0), \\tilde{V}g(u_0)) = 0$ and $|\\det \\left( D_{u=u_0} \\left[ \\nabla g, \\nabla \\tilde{V}g \\right] \\right) | \\leq \\beta$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{lem}[Classification of perturbations of sub-$\\beta$ intersections] \\label{l:class_tang}\nLet $r>0$ and $\\beta>0$ be given. Let $0$ be a regular value of $g_1, g_2 \\in C^{\\infty}({\\mathbb R}^2)$. Furthermore, we assume that $g_1^{-1}(0) \\cap g_2^{-1}(0)$ is regular, i.e. that $|\\det(\\nabla g_1(u), \\nabla g_2(u))| \\neq 0$ for each point $u \\in g_1^{-1}(0) \\cap g_2^{-1}(0)$.  Suppose $u_0 \\in B(u_0, r) \\cap g_1^{-1}(0) \\cap g_2^{-1}(0) \\subset B(R)$ is an isolated point of ``near\"-tangential intersection, that is\n\\begin{equation*}\n|\\det(\\nabla g_1(u_0), \\nabla g_2(u_0))| \\leq \\beta.\n\\end{equation*}\n\nThen there exists $b(r)>0$ with $b < r$, such that for perturbations $\\psi_1, \\psi_2$ with $\\|\\psi_i\\|_{C^2(B(R)} < b$ for $i=1,2$, the only possibilities of zeroes of $(g_1 + \\psi_1)^{-1}(0)  \\cap (g_2 + \\psi_2)^{-1}(0)$ appearing inside of $B(u_0, r + b)$ are: 1) two sub-$(\\beta+b)$ zeroes, that is we have $|\\det(\\nabla (g_1 + \\psi_1), \\nabla (g_2 + \\psi_2))| < \\beta + b$ on $B(u_i, r + b)$ for $i=1,2$ being the new zeroes, or 2) a single sub-$(\\beta + b)$ zero in $B(u_0, r + b)$.\n\\end{lem}\n\n\\begin{proof}\nBy an elementary notion from differential topology \\cite[Chapter 2]{GP}, if $K$ is a compact set and $f: K \\rightarrow {\\mathbb R}^2$ is transverse to a closed submanifold $L \\subsetneq {\\mathbb R}^2$, then the transversality is stable under suitable perturbations (i.e. of class $C^1$) of $f$.\nConsidering this fact, let $f$ be the parametrization of the portion of the nodal component of $g_1^{-1}(0)$ inside of $B(u_0, r)$ and $L$ the nodal component of $g_2^{-1}(0)$ inside of $B(u_0,r)$. If our initial choice of $b$ is small enough, then the transversality will persist. For a possibly smaller choice of $b$ depending on $t$, given value of $r$, this then yields 1) or 2).\n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{thm:euclid_count}}\n\n\\subsection{Some preliminaries}\n\n\\begin{theo}[Grenander-Fomin-Maruyama] \\label{thm:GFM}\nLet $F$ be a random stationary Gaussian field with spectral measure $\\rho$. Then, if $\\rho$ contains no atoms, the action of the translations group\n\\begin{equation*}\n(\\tau_uF)(x) = F(x - u)\n\\end{equation*}\nis ergodic ($F$ is ``ergodic\").\n\\end{theo}\n\n\\begin{theo}[Wiener's Ergodic Theorem] \\label{thm:weiner}\nSuppose that $F$ is ergodic and the translation map ${\\mathbb R}^n \\times C({\\mathbb R}^n) \\rightarrow C({\\mathbb R}^n)$\n\\begin{equation*}\n(u,F) \\mapsto \\tau_u F\n\\end{equation*}\nis measurable with respect to the product $\\sigma$-algebra $\\mathcal{B}({\\mathbb R}^n) \\times \\mathcal{A}$ and $\\mathcal{A}$.  Then every random variable $\\Phi(F)$ with finite expectation ${\\mathbf E} \\left[ |\\Phi(F)| \\right] < \\infty$ satisfies\n\\begin{equation*}\n\\lim\\limits_{R \\rightarrow \\infty} \\frac{1}{\\operatorname{Vol} (B(R))} \\int\\limits_{B(R)} \\Phi(\\tau_u F) \\, du \\rightarrow {\\mathbf E} \\left[ \\Phi(F) \\right],\n\\end{equation*}\nboth almost surely and in $L^1$.\n\\end{theo}\n\n\n\\begin{defn}\n\\label{def:counts}\nLet $\\Gamma \\subset {\\mathbb R}^2$ be a smooth curve (connected or not) and $\\zeta$ be a fixed direction. For $u \\in {\\mathbb R}^2$, $r >0$, and $k \\ge 0$, we denote by\n\\begin{equation*}\n\\pazocal{N}_{\\zeta}(\\Gamma, k, u, r)\n\\end{equation*}\nthe number of connected components of $\\Gamma$ whose number of $\\zeta$-tangencies, that are also lying entirely inside of the ball $B(u,r)$, is precisely $k$.  In the case of $u=0$, we simply write $\\pazocal{N}_{\\zeta}(\\Gamma, k, r)$.  Similarly, we define\n\\begin{equation*}\n\\pazocal{N}^*_{\\zeta}(\\Gamma, k, u, r)\n\\end{equation*}\nby relaxing the condition that the components of $\\Gamma$ merely intersect $B(u,r)$.\n\\end{defn}\n\n\\begin{lem}[Integral-Geometric Sandwich] \\label{l:integro_geom_sandwich}\nLet $\\Gamma$ be a smooth, closed curve and $\\zeta$ be a fixed direction. Then for $0 < r < R$, $k \\ge 0$, we have that\n\\begin{align*}\n\\frac{1}{\\operatorname{Vol}(B(r))} \\int\\limits_{B(R-r)} \\pazocal{N}_{\\zeta}(\\Gamma, k, u, r) \\, du &\\leq \\pazocal{N}_{\\zeta}(\\Gamma, k, R) \\\\\n& \\leq \\frac{1}{\\operatorname{Vol}(B(r))} \\int\\limits_{B(R+r)} \\pazocal{N}^*_{\\zeta}(\\Gamma, k, u, r) \\, du\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}\nOur proof follows almost exactly the steps of \\cite[Lemma 1]{So12} and \\cite[Lemma 5]{BW17}. Here, we present only the proof for the upper bound of $\\pazocal{N}_{\\zeta}(\\Gamma, k, R)$.\n\nLet $\\gamma \\subset \\Gamma$ be a connected component.  Define\n\\begin{equation*}\nG_*(\\gamma) = \\bigcap_{v \\in \\gamma} B(v,r) = \\{ u \\, : \\, \\gamma \\subset B(u,r) \\}\n\\end{equation*}\nand\n\\begin{equation*}\nG^*(\\gamma) = \\bigcup_{v \\in \\gamma} B(v,r) = \\{ u \\, : \\, \\gamma \\cap B(u,r) \\neq \\emptyset \\}.\n\\end{equation*}\nIt follows by the above definition that for all $v \\in \\gamma$,\n\\begin{equation*}\nG_*(\\gamma) \\subset B(v,r) \\subset G^*(\\gamma),\n\\end{equation*}\nwhich in turn immediately yields the inequality\n\\begin{equation*}\n\\operatorname{Vol}(G_*(\\gamma)) \\leq \\operatorname{Vol}(B(v,r)) \\leq \\operatorname{Vol}(G^*(\\gamma))\n\\end{equation*}\nFinally, a sum over the components $\\gamma \\subset B(R)$ whose number of $\\zeta$-tangencies is precisely k gives us\n\\begin{align*}\n\\sum_{\\left\\{\\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}\\right\\}} \\operatorname{Vol}(G_*(\\gamma)) & \\leq \\operatorname{Vol}(B(r)) \\, \\pazocal{N}_{\\zeta}(\\Gamma,k, R) \\\\\n& \\leq \\sum_{\\left\\{ \\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}\\right\\}} \\operatorname{Vol}(G^*(\\gamma))\n\\end{align*}\n\n\nUsing the definition of $G_*(\\gamma)$, we know that\n\\begin{equation*}\n\\sum_{\\left\\{\\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}\\right\\}} \\operatorname{Vol}(G_*(\\gamma)) = \\sum_{\\left\\{ \\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}\\right\\}}\\left( \\int\\limits_{\\{u: \\gamma \\subset B(u,r)\\}} \\, du \\right).\n\\end{equation*}\n\nNote that if $u \\in B(R-r)$ then we have the inclusion $B(u,r) \\subset B(R)$. Hence we have\n\\begin{align*}\n&\\sum_{\\substack{\\gamma \\subset B(R), \\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies}}} \\operatorname{Vol}(G_*(\\gamma)) \\geq \\\\\n&\\int\\limits_{B(R-r)} \\left( \\sum_{\\substack{ \\gamma \\subset B(u,r)\\\\ \\gamma \\mbox{ has } k \\mbox{ tangencies} }} \\right) \\, du = \\int\\limits_{B(R-r)} \\pazocal{N}_{\\zeta}(\\Gamma, k, u, r) \\, du.\n\\end{align*}\nA division by $\\operatorname{Vol}(B(r))$ gives us the desired lower bound. A somewhat similar argument gives the proposed upper bound but we leave this to the reader.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:euclid_count} (1)-(2): existence of an asymptotic law}\n\n\n\\begin{proof}\n\nWe start with $r>0$ fixed and $k \\ge 0$ given.\nDenote $\\pazocal{N}_{\\zeta}(F, k, u, r)$ to be the number of nodal components of $F$ contained in $B(u,r)$ with $k$ $\\zeta$-tangencies,\nand $\\widetilde{\\pazocal{N}}(F,u, r)$ the (total) number of nodal components of $F$ intersecting $\\partial B(u,r)$ (cf. Definition \\ref{def:counts}).\nAn application of Lemma \\ref{l:integro_geom_sandwich} with $\\Gamma=F^{-1}(0)$ and $R$ sufficiently large so that $(1\\pm r\/R)^{2}$ is $\\epsilon$-close to $1$, and bearing in mind the obvious estimate $$\\pazocal{N}^{*}_{\\zeta}(F, k, u, r)\\le \\pazocal{N}_{\\zeta}(F, k, u, r) + \\widetilde{\\pazocal{N}}(F,u, r),$$\nyields the inequality\n\\begin{equation}\n\\label{eq:IGS apply}\n\\begin{split}\n&(1-\\epsilon)\\cdot\\frac{1}{\\operatorname{Vol}(B(R-r))}\n\\int\\limits_{B(R-r)} \\frac{\\pazocal{N}_{\\zeta}(F, k, u, r)}{\\operatorname{Vol}(B(r))} \\, du \\leq  \\frac{\\pazocal{N}_{\\zeta}(F, k, R)}{\\operatorname{Vol}(B(R))} \\\\\n&\\le  (1+\\epsilon)\\cdot\\frac{1}{\\operatorname{Vol}( B(R+r))} \\int\\limits_{B(R+r)} \\frac{\\pazocal{N}_{\\zeta}(F, k, u, r) + \\widetilde{\\pazocal{N}}(F,u, r)}{\\operatorname{Vol}(B(r))}.\n\\end{split}\n\\end{equation}\nNote also that, under the notation of the ergodic Theorem \\ref{thm:weiner}, we have the identity\n$$\\pazocal{N}_{\\zeta}(F, k, u, r)=\\pazocal{N}_{\\zeta}(\\tau_{u}F, k,r)$$ and $\\widetilde{\\pazocal{N}}(F,u, r) = \\widetilde{\\pazocal{N}}(\\tau_{u}F,r)$,\nso that \\eqref{eq:IGS apply} reads\n\\begin{equation}\n\\label{eq:IGS apply trans}\n\\begin{split}\n&(1-\\epsilon)\\cdot \\int\\limits_{B(R-r)} \\frac{\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, r)}{\\operatorname{Vol}(B(r))} \\, du \\leq  \\frac{\\pazocal{N}_{\\zeta}(F, k, R)}{\\operatorname{Vol}(B(R))} \\\\\n& \\le (1+\\epsilon)\\cdot\\frac{1}{\\operatorname{Vol}( B(R+r))} \\int\\limits_{B(R+r)} \\frac{\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, r) + \\widetilde{\\pazocal{N}}(\\tau_{u}F, r)}{\\operatorname{Vol}(B(r))}.\n\\end{split}\n\\end{equation}\n\nAn application of Theorem \\ref{thm:weiner} on the random variable $$\\Phi_{r;\\zeta,k}(F):=\\frac{\\pazocal{N}_{\\zeta}(F, k, r)}{\\operatorname{Vol}(B(R))},$$ of finite expectation in light of Lemma \\ref{lem:Kac Rice Euclid} (1), yields that both\n\\begin{equation}\n\\label{eq:int mean -> L1 Crzk}\n\\frac{1}{\\operatorname{Vol}(B(R-r))}\\int\\limits_{B(R-r)} \\frac{\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, r)}{\\operatorname{Vol}(B(r))} ,\\,\n\\frac{1}{\\operatorname{Vol}( B(R+r))} \\int\\limits_{B(R+r)} \\frac{\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, u, r)}{\\operatorname{Vol}(B(r))} \\xrightarrow[L^{1}]{} \\widetilde{C}_{r;\\zeta,k}\n\\end{equation}\nconverge {\\em in mean} to $$\\widetilde{C}_{r;\\zeta,k}=\\widetilde{C}_{F,r;\\zeta,k}:= \\frac{{\\mathbf E}[\\pazocal{N}_{\\zeta}(\\tau_{u} F, k, u, r)]}{\\operatorname{Vol}(B(r))} \\ge 0 .$$\nSame argument, now employing\nLemma \\ref{lem:Kac Rice Euclid} (2) yields that\n\\begin{equation}\n\\label{eq:intrsct->a}\n\\frac{1}{\\operatorname{Vol}( B(R+r))} \\int\\limits_{B(R+r)} \\frac{\\pazocal{N}(\\tau_{u} F, r)}{\\operatorname{Vol}(B(r))} \\xrightarrow[L^{1}]{} a_{F,r},\n\\end{equation}\nwith\n\\begin{equation}\n\\label{eq:a const=O(1\/r)}\na_{F,r} := \\frac{{\\mathbf E}[\\widetilde{\\pazocal{N}}(\\tau_{u}F,r)]}{\\operatorname{Vol}(B(r))} = \\mathcal{O}_{r\\rightarrow\\infty}\\left(\\frac{1}{r} \\right).\n\\end{equation}\n\nSubstituting \\eqref{eq:a const=O(1\/r)} into \\eqref{eq:intrsct->a}, and then together with \\eqref{eq:int mean -> L1 Crzk} into \\eqref{eq:IGS apply trans} implies that\n\\begin{equation}\n\\label{eq:lim const Cauchy}\n{\\mathbf E}\\left[\\left| \\frac{\\pazocal{N}_{\\zeta}(F, k, R)}{\\operatorname{Vol}(B(R))} -  \\widetilde{C}_{r;\\zeta,k}  \\right|  \\right]= \\mathcal{O}\\left( \\epsilon+\\frac{1}{r}  \\right).\n\\end{equation}\nFinally, taking $r\\rightarrow\\infty$ in \\eqref{eq:lim const Cauchy} implies the existence of the limit\n$$\\widetilde{C}_{k,\\zeta}:=\\lim\\limits_{r\\rightarrow\\infty}\\widetilde{C}_{r;\\zeta,k}.$$\nNow, recalling the Nazarov-Sodin constant $c(\\rho)$ satisfying the defining property \\eqref{eq:NS const def prop},\nyields the convergence \\eqref{eq:conv L1 Euclid}\nto the constants $C_{k,\\zeta}:=\\frac{\\widetilde{C}_{k,\\zeta}}{c(\\rho)}$, at the prescribed rate of $\\mathcal{O}(1\/r)$ of convergence.\nFinally, if $k=0$ or $k$ is odd, then a.s. $\\pazocal{N}_{\\zeta}(F, k, R)=0$ by Lemma \\ref{l:at_least_two_tangencies}, Corollary \\ref{cor:tangencies even} and the classical Bulinskaya Lemma \\cite[Proposition 6.11]{AW}, stronger than merely $C_{k,\\zeta}=0$.\nThis concludes the proof of (1) of Theorem \\ref{thm:euclid_count}.\n\nIn case $F$ is isotropic, that $C_{k,\\zeta}$ are independent of $\\zeta$ follows directly\nfrom the invariance of the law of $F$ w.r.t. the rotations. Alternatively, one can note that, in this case,\nthe constants $C_{r;\\zeta,k}$ as in \\eqref{eq:int mean -> L1 Crzk}, a priori\ndependent on direction $\\zeta$ are, in fact, independent of $\\zeta$ thanks to the invariance of $F$ w.r.t. the rotations of ${\\mathbb R}^{2}$,\nand so are the constants $C_{k,\\zeta}$, which is (2) of Theorem \\ref{thm:euclid_count}.\n\n\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:euclid_count} (3): support of limit direction distribution}\n\nThe deterministic intersection results in \\S\\ref{s:deterministic} play a crucial role in this section. We remind ourselves that for $k$ odd, $C_{k,\\zeta}=0$ follows immediately from the stronger property that $\\pazocal{N}_{\\zeta}(F, k, R)=0$ almost surely.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:euclid_count} (3)]\n\nWe distinguish between two main cases: when $(\\rho 4^{*})$ is satisfied (i.e. the interior of the support of $\\rho$ is non-empty), and when $\\rho=\\sigma_{S^{1}}$.\n\n{\\bf Case $(\\rho 4^{*})$ is satisfied.}\nThe proof follows along similar lines to those of \\cite[Theorem 6]{BW17} and \\cite[Proposition 5.2]{SW18}, namely the ``barrier method\" as initiated in \\cite{NS09} and later generalized in \\cite{So12}.\nThe three main steps are standard, and we summarize them as follows: 1) find a deterministic function on $B(r_0)$, for some $r_0>0$ sufficiently large, so that it contains at least one component with the desired number of tangencies, 2) show that the probability of selecting a random function with almost the same nodal component is positive by establishing that $\\mathcal{H}(\\rho) \\subset C^m(B(R))$ is dense and then using the structure of the Gaussian measure on our probability space $\\Omega$, and 3) assuming the previous two points, show that the expected density of such nodal components is positive via a simple monotonicity argument originating from \\cite{NS09}. Here we give only the details that deviate from referenced article at critical junctures and leave the remaining details to the interested reader.\n\nRecall that by the virtue of axiom $(\\rho 4^{*})$, the interior of supp $\\rho$ is non-empty. This implies, via the same arguments using the existence of approximations to finite sums of Dirac $\\delta$-functions to generate the set of all monomials as in \\cite[Section 3.4]{BW17} that the Hilbert space $\\mathcal{H}(\\rho)$ is dense in $C^m(Q)$, for a given compact set $Q \\subset B(R)$ and for all $m$.\n\nLet $\\zeta$ be a given direction, and $g_k$ be a $C^2({\\mathbb R}^{2})$ function with a regular nodal component $\\Gamma$ that contains the origin, has precisely $k$ tangencies to with respect to the constant vector field $\\zeta$ with each tangency being quantitatively transverse, and is contained in $B(r_0)$. Moreover, we know by the deterministic analysis in \\S\\ref{s:deterministic}, we know there exists a $\\beta_0>0$ such that each point of tangency is $\\beta_0$-transverse for some $\\beta_0>0$ and that each other point of $\\Gamma$ is not sub-$\\beta_0$. By the density of $\\mathcal{H}(\\rho) \\subset C^k(B(R))$, there exists $\\mathfrak{g}_k \\in \\mathcal{H}(\\rho)$ such that $\\|\\mathfrak{g}_k - g_k \\|_{C^2(B(R))} < \\beta_0\/4$.  Furthermore, we find that the neighborhood defined by $\\|\\mathfrak{g} - \\mathfrak{g}_k \\|_{C^2(B(R))} < \\beta_0\/4$ has only random functions $\\mathfrak{g}$ with a nodal component encompassing the origin and with exactly $k$ $\\zeta$-tangencies.  The positivity of the quantity $\\mathbb{P} \\left\\{ \\|\\mathfrak{g} - \\mathfrak{g}_k \\|_{C^2(B(R))} < \\beta_0\/4 \\right\\}$ and consequently of $C_{k, \\zeta}$ are standard; see \\cite{NS15, SW18, BW17}.\n\n{\\bf Case $\\rho=\\sigma_{S^{1}}$.}\nOnce again, our proof follows similar lines as that of \\cite[Theorem 7]{BW17} and \\cite[Proposition 5.2]{SW18} in the case of $\\alpha =1$ except that now we utilize the stronger $C^k(B(R))$ versions of both Whitney's approximation\/extension theorem and the Lax-Malgrange extenstion theorem; see \\cite[Theorem 2.6.3]{Horm} and references therein, and respectively \\cite[Lemma 7.2]{EPS}, for the exact statements. For our purposes, $k=2$ is sufficient.\n\nLet $\\gamma$ be a smooth closed curve in ${\\mathbb R}^2$ that has precisely $k$ (even) tangencies, each of which is quantitatively transverse, with respect to $\\zeta$: that is, we know there exists a $\\beta_0 >0$ such that each of its tangencies is $\\beta_0$-transverse and each other point of $\\gamma$ is not sub-$\\beta_0$. Via an application of the Implicit Function Theorem, we can find a tubular neighborhood $V_{\\gamma}$ of $\\gamma$ and a smooth function $H_{\\gamma}: V_{\\gamma} \\rightarrow {\\mathbb R}$ with $\\gamma = H^{-1}(0)$; furthermore, we can assume that $\\inf_{u \\in V_{\\gamma}} \\| \\nabla H_{\\gamma}(u) \\| > 0$.  Now, the $C^2(B(R))$ version of Whitney's theorem states that for a given $\\epsilon >0$, there exists a real analytic function $G$ such that $\\| G - H_{\\gamma} \\|_{C^2(V_{\\gamma})} < \\epsilon$.\n\nConsider the range $\\beta_0 > \\epsilon > 0$.  By Proposition \\ref{p:stab_trans}, for $\\epsilon$ sufficiently small, we know that $\\widetilde{\\gamma} := G^{-1}(0) \\cap V_{\\gamma}$ has the property that $\\mbox{dist}(\\widetilde{\\gamma}, \\gamma) < \\epsilon$ (this can also be seen from $C^2(B(R))$ version of Thom's Isotopy Theorem; see \\cite{AR}).  Moreover we have that $\\widetilde{\\gamma}$ also has precisely $k$ tangencies with respect to $\\zeta$ each of which is $\\beta_0 - \\epsilon$-transverse. Now, it remains to find a solution of $\\Delta + 1$ that captures this analytic curve $\\widetilde{\\gamma}$ as part of its nodal set.\n\nRepeating the argument as described in the proof of \\cite[Theorem 1.1]{CanSar18}, we use that $\\widetilde{\\gamma}$ separates ${\\mathbb R}^2$ into two components, a bounded component $A_{\\widetilde{\\gamma}}$ and an unbounded component. Let $\\lambda^2$ be the first Dirichlet eigenvalue for the domain $A_{\\widetilde{\\gamma}}$ and let $h_{\\lambda}$ be the corresponding eigenfunction: that is, $(\\nabla + \\lambda^2) h_{\\lambda}(u) = 0$ for $x \\in \\overline{A_{\\widetilde{\\gamma}}}$ and $h_{\\lambda \\, | \\widetilde{\\gamma}} = 0$.\n\nLet $\\overline{\\lambda A_{\\widetilde{\\gamma}}} := \\{ x \\in {\\mathbb R}^2: u\/\\lambda \\in A_{\\widetilde{\\gamma}} \\}$ It follows that $h(u) := h_{\\lambda}(u\/\\lambda)$ solves the Dirichlet problem for $(\\Delta + 1)$ on $\\overline{\\lambda A_{\\widetilde{\\gamma}}}$ and can be extended (thanks to analyticity) to an open set $B_{\\widetilde{\\gamma}}$ such that $\\overline{\\lambda A_{\\widetilde{\\gamma}}} \\subset B_{\\widetilde{\\gamma}}$ as well as solving the following boundary value-type problem:\n\\begin{equation*}\n\\begin{cases}\n(\\Delta + 1)h(u) = 0, & x \\in B_{\\widetilde{\\gamma}} \\\\\nh(u) = 0, & x \\in \\lambda \\widetilde{\\gamma}\n\\end{cases}\n\\end{equation*}\nwhere $\\lambda \\widetilde{\\gamma} := \\{ x \\in {\\mathbb R}^2: u\/\\lambda \\in \\widetilde{\\gamma} \\}$. It is important to notice that $\\lambda \\widetilde{\\gamma}$ continues to have precisely $k$ $\\frac{(\\beta_0-\\epsilon)}{\\lambda^3}$-transverse tangencies with respect to $\\zeta$ thanks to the fact that dilation by $\\lambda$ commutes with the action generated by the constant vector field $\\zeta$ which is translation in the direction $\\zeta$. Added: Now, let $\\epsilon' < \\frac{\\beta_0 - \\epsilon}{10\\lambda^3}$.\n\nOur next-to-penultimate step is an application of a $C^2(B(R))$ version of Lax-Malgrange \\cite[Lemma 7.2]{EPS} which states if $B_{\\widetilde{\\gamma}}^{\\complement}$ has no compact components, then there exists a global solution on ${\\mathbb R}^2$ to the equation $(\\Delta + 1)g(u) = 0$ with the property that $\\|g - h \\|_{C^2(B_{\\widetilde{\\gamma}})} < \\epsilon'$. Another application of Proposition \\ref{p:stab_trans} implies that $g^{-1}(0) \\cap V_{\\widetilde{\\gamma}}$ has precisely $k$ (transverse) tangencies with respect to $\\zeta$. The final step involving disc packing is exactly the same as that for disc packing in the case of $\\alpha < 1$.\n\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{thm:euclid_count} (4): direction distributions not leaking mass}\n\n\\label{sec:proof thm euclid no leak}\n\n\\begin{proof}\n\nGiven a direction $\\zeta\\in S^{1}$ and $R>0$, denote $\\mathcal{N}_{\\zeta}(R)$ to be the number of\n$\\zeta$-tangency points of $F^{-1}(0)$ lying inside $B(R)$ (where the corresponding nodal component might and might not\nbe contained in $B(R)$).\nThe Euclidean analogue of \\eqref{eq:tot numb tang=sum mean} is the inequality\n\\begin{equation}\n\\label{eq:sum k*nod(k)<=Ncc}\n\\sum\\limits_{k=1}^{\\infty}k\\cdot \\pazocal{N}_{\\zeta}(F,k,R) \\le \\mathcal{N}_{\\zeta}(F,R),\n\\end{equation}\nas we have to discount for those tangency points that lye inside $B(R)$, but belong to the nodal components of $F$\nmerely intersecting $\\partial B(R)$. Since all the involved quantities are nonnegative, we may take the expectation\nof both sides of \\eqref{eq:sum k*nod(k)<=Ncc} and choose the order of summation and expectation as we please (e.g. by Monotone Convergence Theorem), so that, upon invoking Lemma \\ref{lem:Kac-Rice Euclid tangencies}, we may write\n\\begin{equation*}\n\\sum\\limits_{k=1}^{\\infty}k\\cdot {\\mathbf E}[\\pazocal{N}_{\\zeta}(F,k,R)] \\le {\\mathbf E}[\\mathcal{N}_{\\zeta}(F,R)]=C_{0}\\cdot \\operatorname{Vol}(B(R)),\n\\end{equation*}\nfor some $C_{0}>0$ depending on the law of $F$ only.\nEquivalently,\n\\begin{equation}\n\\label{eq:sum k*E[k]\/E<=C0}\n\\sum\\limits_{k=1}^{\\infty}k\\cdot \\frac{{\\mathbf E}[\\pazocal{N}_{\\zeta}(F,k,R)]}{{\\mathbf E}[\\pazocal{N}(F,R)]} \\le C_{0}\\cdot \\operatorname{Vol}(B(R))\/{\\mathbf E}[\\pazocal{N}(F,R)] \\le C_{1},\n\\end{equation}\nwith some $C_{1}>0$, holding for $R$ sufficiently large, by \\eqref{eq:NS const def prop}.\n\n\\vspace{2mm}\n\nWe interpret the l.h.s. of \\eqref{eq:sum k*E[k]\/E<=C0} as the average number of $\\zeta$-tangent points per nodal component of $F$,\nimportantly, bounded by the r.h.s. of \\eqref{eq:sum k*E[k]\/E<=C0} for every $R>0$ sufficiently large.\nThe sequence\n\\begin{equation}\n\\label{eq:aRk def}\na_{R;k}:=\\frac{{\\mathbf E}[\\pazocal{N}_{\\zeta}(F,k,R)]}{{\\mathbf E}[\\pazocal{N}(F,R)]}\n\\end{equation}\nsatisfies for every $R>0$ the equality\n\\begin{equation}\n\\label{eq:sum(aRk)=1}\n\\sum\\limits_{k= 0}^{\\infty}a_{R;k}=1,\n\\end{equation}\nand also, for every $k\\ge 0$,\n\\begin{equation}\n\\label{eq:lim(aRk)=Ckz}\n\\lim\\limits_{R\\rightarrow\\infty}a_{R;k}=C_{k,\\zeta},\n\\end{equation}\nthanks to Theorem \\ref{thm:euclid_count} (1), and \\eqref{eq:NS const def prop}. We claim that \\eqref{eq:sum(aRk)=1}\ntogether with \\eqref{eq:lim(aRk)=Ckz} yield\n\\begin{equation}\n\\label{eq:sum(Ck=1)}\n\\sum\\limits_{k\\ge 0}C_{k,\\zeta}=1,\n\\end{equation}\nwhich is the statement of Theorem \\ref{thm:euclid_count} (4). To this end, it is sufficient to prove the {\\em tightness}\nof $\\{a_{R;k}\\}$ w.r.t. $k$, i.e. that for every $\\epsilon>0$ there exists $K_{0}=K_{0}(\\epsilon)$ and $R_{0}=R_{0}(\\epsilon)>0$ sufficiently large,\nso that\n\\begin{equation*}\n\\sum\\limits_{k=K_{0}}^{\\infty}a_{R;k}<\\epsilon\n\\end{equation*}\nholds for all $R>R_{0}$.\nHowever the said tightness condition follows directly from \\eqref{eq:sum k*E[k]\/E<=C0} (on recalling that $a_{R;k}$ are given by\n\\eqref{eq:aRk def}).\nAs mentioned above, this is sufficient for \\eqref{eq:sum(Ck=1)}, which, in turn, is the statement of Theorem \\ref{thm:euclid_count} (4).\n\n\n\\end{proof}\n\n\n\\section{Riemannian scenario: Local results}\n\n\\subsection{Local setting and statement of the main local result}\n\nLet $V \\in \\mathcal{V}(\\M)$, and define\n\\begin{equation*}\n\\pazocal{N}_{V}(f_{T, \\alpha}, k, x, r)\n\\end{equation*}\nto be the number of components of $f_{T, \\alpha}^{-1}(0)$ completely contained in the geodesic ball $B(x, r)$ whose number of $V$-tangencies (not counting the zeroes of $V$ that might touch $f_{T, \\alpha}^{-1}(0)$) is precisely $k$. That is to say, we are counting the number of  components of $f_{T, \\alpha}^{-1}(0)$ completely contained in the geodesic ball $B(x, r)$ with precisely $k$ zeroes of $V f_T$ each.\n\nRecalling the scaled random fields \\eqref{eq:fxT scaled def} and Definition \\ref{d:blown_up_V} of $\\tilde{V}_{x,T}$, the analogous local quantity for $x \\in \\M \\setminus \\{V=0\\}$ is\n\\begin{equation*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R),\n\\end{equation*}\nwhich, by definition, is equal to the number of components of $f_{x,T}^{-1}(0)$ completely contained in the Euclidean ball $B(0, R)$ whose number of $\\tilde{V}_{x,T}$-tangencies is precisely $k$. Equivalently, $\\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R)$ is the number of connected components of\n$f_{x, T}^{-1}(0)$ completely contained in $B(0, R)$ with precisely $k$ zeroes of $\\tilde{V}_{x,T} f_{x,T}$ each.\n\n\\begin{theo} \\label{t:conv_prob}\nLet $f_{T, \\alpha}$ be the random band-limited functions \\eqref{eq:fT band lim alpha<1} (or \\eqref{eq:fT band lim alpha=1}),\nand $V \\in \\mathcal{V}(\\M)$. Then for all $x \\in \\M \\setminus \\{ V=0 \\}$, $k \\ge 0$, and $\\epsilon>0$, we have\n\\begin{equation}\n\\label{eq:double lim R,T conv loc}\n\\lim\\limits_{R \\rightarrow \\infty} \\limsup\\limits_{T \\rightarrow \\infty} \\, \\mathbb{P} \\left[ \\left| \\frac{\\pazocal{N}_{V}(f_T, k, x, R\/T)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R))}  - C_{k} \\right| > \\epsilon \\right] = 0\n\\end{equation}\nwhere the constants $C_{k}$ are same as in Theorem \\ref{thm:euclid_count} (1)\napplied on the random field $F=\\mathfrak{g}_{\\alpha}$ with $\\zeta$ arbitrary\n($C_{k}$ are independent of $\\zeta$ by Theorem \\ref{thm:euclid_count} (2)), and $c_{2,\\alpha}>0$ is the Nazarov-Sodin constant\nof $\\mathfrak{g}_{\\alpha}$, i.e. satisfying the defining property \\eqref{eq:NS const def prop} with $F=\\mathfrak{g}_{\\alpha}$.\n\\end{theo}\n\n\\subsection{Exceptional events} \\label{s:exceptional_events}\n\nIn this section, we introduce the following parameters and their purposes: a small parameter $\\delta>0$ to control probabilities, $b>0$ to control the quality of coupling approximation, $\\beta_{1,i} > 0$ to control the regularity of our components of $\\mathfrak{g}_{\\alpha}$ for $i=1,2$, $\\beta_{2,i} >0$ to control the nature of our tangencies for $i=1,2$, $M_i$ to control the $C^k$ norms of our random functions for $i=0,1$, large spectral parameter $T>0$, and related $R>0$ controlling our convergence to the scaling limit.\n\n\\begin{defn}[Exceptional events] \\label{d:standard_events}\nLet $R,T, b, M_0, M_1, \\beta_1, \\beta_2 >0$ and $x \\in \\M \\setminus \\{ V=0 \\}$. We define the events as follows:\n\\begin{align*}\n& \\Delta_1 = \\Delta_1(x, R,T;b) = \\{ \\|f_{x,T} - \\mathfrak{g}_{\\alpha} \\|_{C^2(B(2R))} \\geq b \\} \\\\\n& \\Delta_2 = \\Delta_2(R, M_0) = \\{ \\| \\mathfrak{g}_{\\alpha} \\|_{C^2(B(2R))} \\geq M_0 \\} \\\\\n& \\Delta_3 = \\Delta_3(x, R, T, M_1) = \\{ \\| f_{x,T} \\|_{C^2(B(2R))} \\geq M_1 \\} \\\\\n& \\Delta_4 = \\Delta_4(R, \\beta_{1,1}) = \\{ \\min_{u \\in B(2R)} \\max \\{ |\\mathfrak{g}_{\\alpha}(u)|, \\|\\nabla \\mathfrak{g}_{\\alpha}(u)\\|_{2}  \\} \\leq \\beta_{1,1} \\} \\\\\n& \\Delta_5 = \\Delta_5(x, R, T, \\beta_{1,2}) = \\{ \\min_{u \\in B(2R)} \\max \\{ |\\tilde{V}_{x,T}(\\mathfrak{g}_{\\alpha})(u)|, \\|\\nabla (\\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha})(u)\\|_{2}  \\} \\leq \\beta_{1,2} \\}.\n\\end{align*}\n\\end{defn}\n\nWe would like to emphasize that for the event $\\Delta_1$, its analogue in \\cite[Lemma 4]{So12}  only involves the $C^1$ norm and is stated to have a quantitatively low probability. It is however clear, thanks to the flexibility of the Hadamard-Landau inequality, that we can immediately replace $C^1$ by $C^2$ both in the statement of \\cite[Lemma 4]{So12} and its proof. The parameter $\\beta_{1,2}$ in some sense measures our distance away from the set $\\{ V = 0\\}$ on $\\M$.\n\n\\begin{defn} \\label{d:unif_stability_fields}\n For $R>0$, and $\\eta, \\beta_{1,3}>0$ small, we define the \\textit{unstable components} event $\\Delta_6(\\eta, R, \\beta_{1,3})$ as follows:\n\\begin{align*}\n\\Delta_6(\\eta, R,\\beta_{1,3}) := &\\{ \\mbox{the number of components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ with a point $u_0$} \\\\\n& \\mbox{such that } | \\nabla \\mathfrak{g}_{\\alpha}(u_0)| \\leq  \\beta_{1,3} \\mbox{ is at least $\\eta R^2$} \\}\n\\end{align*}\n\\end{defn}\n\n\\begin{defn}  \\label{d:unstable_event}\nFor $x \\in \\M\\setminus \\{V = 0\\}$, $R, T>0$, and $\\eta, \\beta_2>0$ small, we define the \\textit{tangentially unstable components} event $\\Delta_7(V, x, R, T, \\beta_2, \\eta)$ as follows:\n\\begin{align*}\n\\Delta_7(V, x, R, T, \\beta_2, \\eta) := &\\{ \\mbox{the number of components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$, with point $u_0$ such that} \\\\\n& \\det [\\nabla \\mathfrak{g}_{\\alpha}(u_0),  \\nabla \\left( \\tilde{V}_{x,T} \\, \\mathfrak{g}_{\\alpha} \\right) (u_0) ] | \\leq  \\beta_2 \\mbox{, is at least $\\eta R^2$} \\}\n\\end{align*}\nwhere $\\tilde{V}_{x,T}$ is the blown-up vector field at $x$, at scale $T$, as given in Definition \\ref{d:blown_up_V}.\n\\end{defn}\n\nA combination of the Borel-TIS and Sudakov-Fernique inequalities along with the argument in \\cite[Section 4.3]{BW17} using Chebyshev's Inequality give us a probability of less than $\\delta$ for the events $\\Delta_2$ and $\\Delta_3$.\n\n\\begin{lem}[Uniform stability for components of smooth Gaussian fields, Cf. {\\cite[Proposition 4.3]{SW18}}] \\label{p:unstable_comp}\nGiven $\\delta>0$, $\\eta >0$, there exists $\\beta_{1,3}(\\delta,\\eta)>0$ (possibly small) and $R_0(\\delta, \\eta)$ such that for all $R \\geq R_0$, we have that $\\mathbb{P}[\\Delta_6(\\eta, R, \\beta_{1,3})] < \\delta$.\n\\end{lem}\n\n\\begin{prop}[Uniform stability of $\\beta_2$-transverse tangencies for smooth Gaussian fields] \\label{p:unstable_tangencies}\nGiven $\\eta, \\delta > 0$ and $x \\in \\M\\setminus\\{V = 0\\}$, there exists $\\beta_{2}=\\beta_{2}(\\eta,\\delta, V)>0$, and in turn\n$R_0(\\delta, \\eta, \\beta_{2},x)$ such that for all $R \\geq R_0$, there exists a $T_0(R, \\beta_{2}, V)>0$ such that for all $T \\geq T_0$, we have $\\mathbb{P}[\\Delta_7(V,x,R, T, \\beta_2, \\eta)] < \\delta$.\n\\end{prop}\n\n\nWe would like to remind the reader of the following standard fact:\n\\begin{lem} \\label{l:qual_bulinskaya}\nFor all $x \\in \\M \\setminus \\{V=0\\}$, we have that the event\n\\begin{align*}\n\\Delta_8(V, x, T, R) &= \\bigg\\{ \\mbox{there exists } u_0 \\in B(R) \\mbox{ such that } \\mathfrak{g}_{\\alpha}(u_0)=\\left(\\tilde{V}_{x,T} \\, \\mathfrak{g}_{\\alpha} \\right) (u_0) \\\\\n&=  \\det \\left[\\nabla \\mathfrak{g}_{\\alpha}(u_0),  \\nabla \\left( \\tilde{V}_{x,T} \\, \\mathfrak{g}_{\\alpha} \\right)(u_0) \\right] = 0 \\bigg\\}\n\\end{align*}\nhas probability 0.\n\\end{lem}\n\n\n\\begin{proof}[Proof of Lemma \\ref{l:qual_bulinskaya}]\nThis is an immediate application of Bulinskaya's lemma \\cite[Proposition 6.11]{AW} on the random field $(\\mathfrak{g}_{\\alpha}, \\tilde{V}_{x,T} \\mathfrak{g}_{\\alpha})$.\n\\end{proof}\n\nThanks to having both $\\mathfrak{g}_{\\alpha}^{-1}(0)$ and $( \\tilde{V}_{x,T} \\mathfrak{g}_{\\alpha})^{-1}(0)$ (and the zero sets for the corresponding perturbations) being regular with probability 1, that the intersection between these two sets is always transversal by Lemma \\ref{l:qual_bulinskaya}, and finally Lemma \\ref{l:even_int}, we are able to focus only on events where the nodal components have an even number of tangencies to $\\tilde{V}_{x,T}$ (although $k=0$ may occur and is addressed in the proof of Proposition \\ref{p:stab_k_even}). Thus, we have reduce our analysis to that of $\\beta$-transverse tangencies and sub-$\\beta$ tangencies.\n\nIn closing this, we set\n\\begin{equation}\n\\label{eq:E good event def}\nE = \\cap_{i=1}^8 \\Delta_i^{\\complement}\n\\end{equation}\nwith the corresponding range of parameters so that $\\mathbb{P}[E] > 1 - \\delta$, where $\\delta > 0$ is given. We will refer to this $E$ numerous times in upcoming sections.\n\n\n\\subsection{High probability stability estimates for local counts} \\label{s:stab_est}\n\nA $\\mbox{sub}-\\frac{3}{2} \\beta$ tangency of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ is mapped, under a small $C^{2}$ perturbation, to $\\mbox{sub}-\\left( \\frac{3}{2}\\beta + b \\right)$ tangency of $f_{x,T}^{-1}(0)$, where $b$ is small (see Definition \\ref{d:standard_events}).  We will later prove that components containing these near-degenerate tangencies are {\\em few} in number, with high probability (see Proposition \\ref{p:unstable_tangencies}).\n\n\n\\begin{prop}[Stability for the number of tangencies] \\label{p:stab_k_even}\nLet $V \\in \\mathcal{V}(\\M)$ be given, and let $\\tilde{V}_{x,T}$ denote the blow-up of $V$ at $x$. We have that for all $x \\in \\M \\setminus \\{V = 0\\}$, $\\delta>0$, $\\eta >0$, and $k \\ge 0$, there exists a $R_0(x, \\delta, \\eta, \\|V(x)\\|)$ such that for all $R \\geq R_0$, there exists a $T_0(R, \\|V(x)\\|, \\delta, k)$ such that for all $T \\geq T_0$, we have the estimate\n\\begin{align}\n\\label{eq:perturb count}\n \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k , R-1) - \\eta R^2 & \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k , R)\n \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k , R+1) +  \\eta R^2,\n\\end{align}\nwith probability $> 1 - \\delta.$\n\\end{prop}\n\nNote that in Proposition \\ref{p:stab_k_even} we are counting all $k$ tangencies, whether they are $\\beta$-transversal or sub-$\\beta$. This is the source of the extra terms of $\\pm\\eta R^{2}$ on both l.h.s. and r.h.s. of \\eqref{eq:perturb count}, accounting for the sub-$\\beta$, or rather near-degenerate, tangencies.\n\n\n\nLet $V \\in \\mathcal{V}$. We will apply our deterministic intersection results from \\S\\ref{s:deterministic} to smooth Gaussian fields $G = (\\mathfrak{g}_{\\alpha}, \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha})$, where $\\tilde{V}_{x,T}$ is the blow-up of $V$ at $x \\in \\M$,  and $(f_{x,T}, \\frac{1}{T}(Vf_T)_{x,T})$. It is important to note that there two advantages to considering $\\frac{1}{T}(Vf_T)_{x,T}$. First $(\\frac{1}{T} V)f_T$ and $ V(f_T)$ share the same zero sets and second, the zeroes of $\\frac{1}{T}(Vf_T)_{x,T}$ and $\\tilde{V}_{x,T}(f_{x,T})$ are close with respect to a natural perturbation parameter that appears when summoning certain coupling results (cf. \\cite{SW18, So12} and Definition \\ref{d:standard_events}).\n\n\\begin{lem}[Blowing up ``almost\" commutes with differentiation] \\label{l:non_commuting}\nGiven $x \\in \\M\\setminus\\{V=0\\}$, and $R>0$, there exists $T_0(x,R,b)$ such that for all $T \\geq T_0$, $\\tilde{V}_{x,T}$ is non-vanishing in $B(R) \\subset T_x\\M$ and we have that\n\\begin{align*}\n T \\tilde{V}_{x,T} \\left( f_{x,T} \\right)  = (Vf_T)_{x,T}+ \\mathcal{O}_{\\|f_{x,T}\\|_{C^1}}\\left( \\frac{R}{T}\\right).\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nThis is an immediate consequence of Taylor's theorem and the fact that for $T$ large enough, $R\/T < \\min \\{inj(\\M), 1 \\}$ and that $D(\\exp_x)_{Z = \\frac{Y}{T}} = Id + \\mathcal{O} \\left( \\frac{R}{T} \\right)$ as $|Y|\/T \\leq R\/T$.\n\\end{proof}\n\nWe need an intermediate estimate on the plane before proceeding to the proof of Theorem \\ref{t:conv_prob}:\n\\begin{lem}[Close vector fields generate the same count] \\label{l:close_vector_fields}\nLet $\\delta > 0$ be given.  Consider the parameter $\\beta_2$ that controls degeneracy of our tangencies as described in \\S\\ref{s:exceptional_events}.  Let $0 < \\epsilon \\leq \\beta_2$ be given. Suppose $\\tilde{V}_1, \\tilde{V}_2 \\in \\mathcal{V}({\\mathbb R}^2)$ are non-vanishing and that $\\| \\tilde{V}_1 - \\tilde{V}_2 \\|_{C^1(B(R))} < \\epsilon$; that is, their coefficients with respect to the standard basis are $C^1$ close.\nThen, for all $k \\ge 0$, for all $\\eta >0$, on the event $E$ as in \\eqref{eq:E good event def} (of probability $>1-\\delta$), there exists $R_0(\\eta, \\epsilon)$ such that for all $R \\geq R_0$, the following inequality holds:\n\\begin{align*}\n\\pazocal{N}_{\\tilde{V}_1}(\\mathfrak{g}_{\\alpha}, k , R-1) - \\eta R^2 & \\leq \\pazocal{N}_{\\tilde{V}_2}(\\mathfrak{g}_{\\alpha}, k , R) \\leq  \\pazocal{N}_{\\tilde{V}_1}(\\mathfrak{g}_{\\alpha}, k , R+1) + \\eta R^2.\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}[Proof of Lemma \\ref{l:close_vector_fields}]\nThe proof follows almost every step of the proof of our main Proposition \\ref{p:stab_k_even} given in \\S\\ref{sect:proof_main_stab_prop} except that we do not perturb by the vector function $( f_{x,T} - \\mathfrak{g}_{\\alpha}, \\tilde{V}_{x,T}(f_{x, T}  - \\mathfrak{g}_{\\alpha}) )$ but by the simpler function $(0, (V_2-V_1) \\mathfrak{g}_{\\alpha})$. This type of perturbation is under the scope of the hypotheses of Lemma \\ref{l:Sodin_type_calc_lem} where we set $b < \\epsilon$ and $\\Psi = (0, (V_2-V_1) \\mathfrak{g}_{\\alpha})$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{t:conv_prob} assuming Proposition \\ref{p:stab_k_even}, Lemma \\ref{l:non_commuting} and Lemma \\ref{l:close_vector_fields}]\n\nConsidering the listed proposition and lemmas, we let $\\delta, \\eta>0$ be given and $k\\ge 0$. Now take the parameters as prescribed in \\S\\ref{s:exceptional_events}: that is, take $R_0, M_0, M_1, \\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2$, and $T_0$.\n\nWe assume that\n\\begin{equation} \\label{e:final_freq_stab}\nR\/T < \\min \\left\\{1, \\mbox{inj}(\\M), \\frac{1}{10 M_0}, \\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2 \\right\\}\n\\end{equation}\nafter possibly taking the initial $T_0$ larger than is posed in \\S\\ref{s:exceptional_events}.  We know that $(Vf_T)_{x,T} = T (\\tilde{V}_{x,T})(f_{x,T}) + \\mathcal{O}_{\\|f_{x,T}\\|_{C^1}}\\left( \\frac{R}{T} \\right)$ in the local coordinates w.r.t. $\\exp_x$ by Lemma \\ref{l:non_commuting}.  Let us define a new quantity $\\pazocal{N} \\left(f_{x,T}, (Vf_T)_{x,T}, k,  R  \\right)$ and set it equal to the number of components of $f_{x,T}^{-1}(0)$ contained inside of $B(R)$ whose number of points of intersection with $(Vf_T)_{x,T}^{-1}(0)$ is exactly $k$.  Recall that in Section 4, we established that understanding the number of tangencies was equivalent to understanding the number of intersections between two sets of curves.  From this, we see immediately that $\\pazocal{N}_{V}(f_T, x, k, \\frac{R}{T})$ is bounded above and below by\n\\begin{equation*}\n\\pazocal{N} \\left(f_{x,T}, (Vf_T)_{x,T}, k,  R \\pm 1  \\right).\n\\end{equation*}\n\nNow by our choice of $T_0$ as indicated after \\eqref{e:final_freq_stab}, Lemma \\ref{l:close_vector_fields}\n(in particular, its proof which uses the equivalence between counting tangencies and counting intersections), and Corollary \\ref{l:Sodin_annuli}, we have that $\\pazocal{N} \\left(f_{x,T}, (Vf_T)_{x,T}, k,  R \\pm 1  \\right)$ itself is bounded above and below\nby $\\pazocal{N}_{T\\tilde{V}_{x,T}}\\left( f_{x,T}, k,  R \\pm 2  \\right) \\pm \\eta R^2.$ Thus,\n\\begin{equation*}\n \\pazocal{N}_{T \\, \\tilde{V}_{x,T}}(f_{x,T}, k, R-2)  - \\eta R^2 \\leq \\pazocal{N}_{V}\\left( f_T,x, k,  \\frac{R}{T}  \\right) \\leq \\pazocal{N}_{T \\, \\tilde{V}_{x,T}}(f_{x,T}, k, R+2)   + \\eta R^2.\n\\end{equation*}\nNow, we emphasis that $\\pazocal{N}_{T \\, \\tilde{V}_{x,T}}(f_{x,T}, k, R \\pm 2) =  \\pazocal{N}_{ \\tilde{V}_{x,T}}(f_{x,T}, k, R \\pm 2)$ since joint zeroes of $(f_{x,T}, T \\, \\tilde{V}_{x,T}(f_{x,T}))$ are the same as those of $(f_{x,T}, \\tilde{V}_{x,T}(f_{x,T}))$, therefore giving us a one-to-one correspondence between components of $f_{x,T}^{-1}(0)$ with precisely $k$ zeroes of $\\tilde{V}_{x,T}(f_{x,T})$ and those with precisely $k$ zeroes of $T\\tilde{V}_{x,T}(f_{x,T})$.  This leads us to\n\\begin{equation*}\n \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R-2)  - \\eta R^2 \\leq \\pazocal{N}_{V}\\left( f_T,x, k,  \\frac{R}{T}  \\right) \\leq \\pazocal{N}_{ \\tilde{V}_{x,T}}(f_{x,T}, k, R+2)   + \\eta R^2.\n\\end{equation*}\n\nIn the local coordinates around $x$ given by $y=\\exp_x(Y)$, we set $\\tilde{V}(x)$ to be the constant vector field given by the trivial extension of $(\\tilde{V}_{x,T})_{|Y=0}$ to ${\\mathbb R}^2$. Proposition \\ref{p:stab_k_even} yields us with probability $1 - \\delta$ that there exists $R_1$ such that $R \\geq R_1(x)$ and $T_1(R)$ such that $T \\geq T_1$, along with an application of the vector field comparison statement Lemma \\ref{l:close_vector_fields} in comparing $\\tilde{V}_{x,T}$ to the constant field $\\tilde{V}(x)$ by taking $T$ as in (\\ref{e:final_freq_stab}), we have\n\\begin{align*}\n \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k , R-1) - 2\\eta R^2 & \\leq \\pazocal{N}_{\\tilde{V}(x)}(f_{x,T},  k , R)  \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k , R+1) + 2\\eta R^2.\n\\end{align*}\nNow, apply Lemma \\ref{l:close_vector_fields} for the second time in the case of $f_{x,T}$, $V_1 = \\tilde{V}_{x,T}$, $V_2 = \\tilde{V}(x)$, and the same values of $R,T$ as in \\eqref{e:final_freq_stab}, to obtain\n\\begin{equation*}\n \\pazocal{N}_{\\tilde{V}(x)}(f_{x,T}, k, R-3)  - \\eta R^2 \\leq \\pazocal{N}_{V}\\left( f_T,x, k,  \\frac{R}{T}  \\right) \\leq \\pazocal{N}_{\\tilde{V}(x)}(f_{x,T}, k, R+3)   + \\eta R^2.\n\\end{equation*}\n\nLet $\\epsilon_1>0$ be given. By the above estimates, occurring with probability $> 1 - \\delta$, this implies\n\\begin{align*}\n\\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{V}\\left( f_T, x,  k , \\frac{R}{T} \\right)}{c_{2, \\alpha} \\pi R^2} - C_k \\right| > \\epsilon_1 \\right] & \\leq\n\\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{\\tilde{V}(x)}(\\mathfrak{g}_{\\alpha}, k , R-4) }{c_{2, \\alpha} \\pi R^2} - C_k \\right| > \\epsilon_1 - \\frac{4\\eta}{c_{2, \\alpha} \\pi} \\right] + \\\\\n& \\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{\\tilde{V}(x)}(\\mathfrak{g}_{\\alpha}, k , R+4) }{c_{2, \\alpha} \\pi R^2} - C_k \\right| > \\epsilon_1 - \\frac{4\\eta}{c_{2, \\alpha} \\pi} \\right] +  \\mathcal{O}(\\delta); \\\\\n\\end{align*}\nnote we have used that $R^2 = (R - 2)^2 + \\mathcal{O}(R)$ in $R$ along with the Kac-Rice formula applied to critical points for $\\mathfrak{g}_{\\alpha}$ as in \\cite[Corollary 2.3]{SW18} to show that\n\\begin{equation*}\n\\frac{ \\pazocal{N}_{\\frac{\\partial}{\\partial Y_1}}(\\mathfrak{g}_{\\alpha}, k , R-4) }{c_{2, \\alpha} \\pi R^2}  = \\frac{ \\pazocal{N}_{\\frac{\\partial}{\\partial Y_1}}(\\mathfrak{g}_{\\alpha}, k , R-4) }{c_{2, \\alpha} \\pi (R-4)^2} + o(1)\n\\end{equation*}\nfor $R \\geq R_3$.  To guarantee that $ \\epsilon_1 - \\frac{4\\eta}{c_{2, \\alpha} \\pi} > \\frac{\\epsilon_1}{2}$, we choose $\\eta$ possibly smaller.\n\nThanks to all of this, we have that for any radius parameter $R \\geq R_4$, we can take $T \\geq T_4$ (for $R_4$ and $T_4$ chosen possibly larger once again after invoking our Euclidean nodal count result Theorem \\ref{thm:euclid_count}) to obtain the estimate\n\\begin{equation*}\n\\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{\\tilde{V}(x)}(\\mathfrak{g}_{\\alpha}, k , R \\pm 4) }{c_{2, \\alpha} \\pi (R \\pm 4)^2} - C_k \\right| > \\frac{\\epsilon_1}{2} \\right] = o(1)\n\\end{equation*}\nwhich in turn implies that\n\\begin{equation*}\n\\mathbb{P}\\left[ \\left| \\frac{ \\pazocal{N}_{V}\\left( f_T, x,  k , \\frac{R}{T} \\right)}{c_{2, \\alpha} \\pi R^2} - C_k \\right| > \\epsilon_1 \\right] = o(1)\n\\end{equation*}\nThis concludes our proof.\n\\end{proof}\n\n\\subsection{Proof of Proposition \\ref{p:stab_k_even}: stability estimate} \\label{sect:proof_main_stab_prop}\n\nIn this section we always take $\\tilde{V} \\in \\mathcal{V}({\\mathbb R}^2)$ to be nowhere vanishing.\n\n\\begin{defn}\n\n\\begin{enumerate}\n\n\\item We denote by $\\pazocal{N}_{\\tilde{V}}^{\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R)$ the number of connected components of\n$\\mathfrak{g}_{\\alpha}^{-1}(0)$, strictly contained in $B(R)$, whose number of $\\tilde{V}$-tangencies is precisely $k$ with each point of tangency being $\\beta$-transverse (cf. Definition \\ref{d:beta_trans}).\n\n\\item We denote by $\\pazocal{N}_{\\tilde{V}}^{\\mbox{sub}-\\beta}(\\mathfrak{g}_{\\alpha}, k, R)$ the number of connected components of\n$\\mathfrak{g}_{\\alpha}^{-1}(0)$, strictly contained in $B(R)$,  whose number of $\\tilde{V}$-tangencies is precisely $k$ with at least one point of tangency being sub-$\\beta$ (cf. Definition \\ref{d:sub_beta}).\n\n\\end{enumerate}\n\n\\end{defn}\n\n\\begin{prop}[Stability of $\\beta$-transversality] \\label{p:b_trans_est}\nLet $k \\ge 0, \\eta, \\delta > 0$, and $x \\in \\M \\setminus \\{V = 0\\}$ be given. Then there exists $\\beta_0(x)$ and $R_0(V,x, \\eta)>0$ such that for all $\\beta < \\beta_0$ and $R \\geq R_0$, there exists $T_0(V,x, R, \\beta, \\eta)>0$ and $b(\\beta, V)>0$ such that for all $T \\geq T_0$, we have the estimate\n\\begin{align*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}^{\\frac{3}{2}\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R-1) & \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\beta-\\mbox{trans}}(f_{x,T}, k, R)\n \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\frac{1}{2}\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R+1)\n\\end{align*}\nfor an event $E$, where $\\mathbb{P} \\left[ E \\right] > 1 - \\delta$ and $\\|f_{x,T} - \\mathfrak{g}_{\\alpha} \\|_{C^2(B(R))} < b$.\n\\end{prop}\n\n\\begin{proof}[Proof of Proposition \\ref{p:stab_k_even} assuming Proposition \\ref{p:b_trans_est} and Proposition \\ref{p:unstable_tangencies}]\nWe prove the first inequality as the second inequality will be proved in an identical fashion. Let $\\eta>0$ be given as in the referenced propositions, and assume that $E$ as in \\eqref{eq:E good event def} occurs.\nSince we have\n\\begin{align*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k, R-1) &= \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\frac{3}{2}\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R-1) + \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\mbox{sub}-\\frac{3}{2}\\beta}(\\mathfrak{g}_{\\alpha}, k, R-1),\n\\end{align*}\nProposition \\ref{p:unstable_tangencies} implies that for $\\beta$ possibly smaller than $\\beta_0(\\eta) = \\beta_2$ as initally given in \\S\\ref{s:exceptional_events}, that\n\\begin{align*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k, R-1) \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\frac{3}{2}\\beta-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R-1) + \\eta R^2\n\\end{align*}\nfor all $R \\geq R_0$ large enough and $T \\geq T_0$ with $T_0$ being large enough and depending on $R$ amongst other parameters.  We are now in a position to apply Proposition \\ref{p:b_trans_est} and repeat this argument for $\\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R)$; that is, we write\n\\begin{align*}\n& \\pazocal{N}_{\\tilde{V}_{x,T}}(\\mathfrak{g}_{\\alpha}, k, R-1) \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\beta-\\mbox{trans}}(f_{x,T}, k, R) + \\eta R^2 \\\\\n& = \\left( \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R) - \\pazocal{N}_{\\tilde{V}_{x,T}}^{\\mbox{sub}-\\beta}(f_{x,T}, k, R) \\right) + \\eta R^2 \\\\\n& \\leq  \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R) + \\eta R^2\n\\end{align*}\nsince $\\pazocal{N}_{\\tilde{V}_{x,T}}^{\\mbox{sub}-\\beta}(f_{x,T}, k, R) \\geq 0$. Note that for the upper bound of $ \\pazocal{N}_{\\tilde{V}_{x,T}}(f_{x,T}, k, R)$, a similar argument follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{p:b_trans_est} assuming the results in \\S\\ref{s:exceptional_events} and \\S\\ref{s:local_results}]\n\nLet $\\eta, \\delta>0$ be given and $\\mathfrak{G} = (\\mathfrak{g}_{\\alpha}, \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha})$. Now, consider the multi-parameter-dependent event $$E(x,R,T, \\eta, \\delta, \\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2, b),$$ as in \\eqref{eq:E good event def}.\nLet $\\beta_0 := \\min \\{  \\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2\\}$ and set $b < \\frac{\\beta_0}{4}$. We now satisfy the hypotheses of Proposition \\ref{p:stab_trans}.  On the corresponding event $E$, we obtain some mapping properties of the components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ to the components of $f_{x,T}$ after perturbations of $C^1$ size $b$. More specifically, we know that after adding the perturbation $f_{x,T} - \\mathfrak{g}_{\\alpha}$ to $\\mathfrak{g}_{\\alpha}$, we have that each of the $\\frac{3}{2} \\beta_2$-transverse tangencies of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ goes to a $ \\left( \\frac{3}{2}\\beta_2 - b \\right)$-transverse tangency of $f_{x,T}^{-1}(0)$. Thus, by Corollary \\ref{l:Sodin_annuli} a component of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ whose number of tangencies is precisely $k$ and each of which is $\\frac{3}{2} \\beta_2$-transverse is mapped to a component of $f_{x,T}^{-1}(0)$ whose number of tangencies is precisely $k$ and each of which is $(\\frac{3}{2} \\beta_2 - b)$- transverse. Thus, we have some form of injectivity and it follows that\n\\begin{equation*}\n\\pazocal{N}_{\\tilde{V}_{x,T}}^{(\\frac{3}{2}\\beta_2 )-\\mbox{trans}}(\\mathfrak{g}_{\\alpha}, k, R-1) \\leq \\pazocal{N}_{\\tilde{V}_{x,T}}^{(\\frac{3}{2}\\beta_2 - b)-\\mbox{trans}}(f_{x,T}, k, R).\n\\end{equation*}\nThe second inequality follows similarly.\n\\end{proof}\n\n\n\\subsection{Proof of Proposition \\ref{p:unstable_tangencies}: high probability of few components with near-degenerate tangencies}\nIn this section, let $T>0$ and $\\mathfrak{G}: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be the smooth Gaussian field\n\\begin{equation}\n\\label{eq:frakG def}\n\\mathfrak{G}(u) := \\left( \\mathfrak{g}_{\\alpha}(u), \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha}(u)\\right).\n\\end{equation}\nIt is important to note that all the results in this section also hold for the Gaussian field $\\mathfrak{F}(u) := \\left( f_{x,T}(u), \\tilde{V}_{x,T}f_{x,T}(u)\\right)$ thanks to the exclusion of the $C^2$ decoupling event $\\Delta_1$ as in \\S\\ref{s:exceptional_events}.\n\n\\begin{defn}\nLet $\\eta>0$, and $\\beta_{2,2}>0$ be small, $R, R_1>0$ be large parameters such that $0< R_1 < R$ and $R\/R_1$ is itself large. Let us cover $B(R)$ with approximately $(R\/R_1)^2$ balls $\\mathcal{D}_i$ of radius $R_1$ such that the multiplicity of the covering is bounded by a constant $\\kappa>0$.  Denote by $\\mathcal{G}_i$ the balls centred at the same points as $\\mathcal{D}_i$ with radii $3R_1$. Note that the multiplicity of the covering $\\{ \\mathcal{G}_i \\}_i$ is bounded by $c_0(2) \\kappa$.\n\n\\begin{enumerate}\n\\item We say that the smooth random field $G=(g_1,g_2)$ is $( \\beta_{2,2}, 3R_1)$-stable on a ball $\\mathcal{G}_i$ if for all $u \\in \\mathcal{G}_i$, we\nhave $|g_1(u)| > \\beta_{2,2}$ or\n$|\\det DG(u)| > \\beta_{2,2}$; otherwise we say that\n$G$ is $(\\beta_{2,2})$-unstable on $\\mathcal{G}_i$.\n\\item We say that $G$ is $(\\eta, \\beta_{2,2}, 3R_1)$-stable if $G$ is $(\\beta_{2,2}, 3R_1)$-stable on all $\\mathcal{G}_i$ except\nfor up to $\\eta R^2$ ones.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{prop} \\label{p:abc_stability}\nLet $x \\in \\M \\setminus \\{V=0\\}$, and $R_{1}>0$. Given $\\delta>0$, $\\eta>0$, there exists a positive number $\\beta_{2,2}(\\delta,R_{1},\\eta)$ so that for all $R>0$ with $R\/R_1 > 100$, there exists a spectral parameter value $T_0(V, R, \\beta_{2,2})$ such that for all $T \\geq T_0$, any random field $\\mathfrak{G}_{|B(R)}$ as in \\eqref{eq:frakG def} is also $(\\eta,\\beta_{2,2}, 3R_1)$-stable\nwith probability $1 - \\delta$.\n\\end{prop}\n\nNote that there is an implicit dependence of all the parameters above on the radius $R_1$ for the elements in our covering. We will track this throughout our calculations. To prove this proposition, we quote the following two lemmas from \\cite{SW18}:\n\n\\begin{lem}[Cf. {\\cite[Lemma 4.9]{SW18}}] \\label{l:4_separation}\nThere exists a constant $c_0 = c_0(\\kappa) > 0$ depending only on $\\kappa$ with the following property: for $\\mathcal{G} = \\{ \\mathcal{G}_i \\}_{i \\leq K}$ a collection of radius-$3R_1$ balls lying in $B(R)$ such that each point $x \\in B(R)$ lies in at most $\\kappa$ elements of $\\mathcal{G}$, we have that $\\mathcal{G}$ contains at most $c_0 \\, K$ balls that are in addition $4$-separated.\n\\end{lem}\n\n\n\\begin{lem}[Cf. {\\cite[Lemma 4.10]{SW18}}] \\label{l:derivative_bounds}\nLet $F$ be a stationary Gaussian random field. For all $\\delta>0$ and for all $K, m \\in \\mathbb{N}$, there exists $C_0(\\epsilon) >0$ such that for any (possibly random) collection of centers $\\{ u_i \\}_{i \\leq K}$ satisfying $d(u_i, u_j) > 4$ for $i \\neq j$, there exist $\\lfloor K\/2 \\rfloor$ points $\\{ u_{i_j} \\}_{i_j \\in I}$ with $|I| = \\lfloor K\/2 \\rfloor$ such that\n\\begin{equation} \\label{e:small_derivatives}\n\\sup\\limits_{| \\nu | \\leq m, B(u_{i_j}, 1)} | \\partial^{\\nu} F(u) | \\leq C_0 \\frac{R}{\\sqrt{K}}\n\\end{equation}\nholds with probability $> 1 - \\delta$.\n\\end{lem}\n\nFor the field $F = \\mathfrak{g}_{\\alpha}$, the constant $C_0$ in (\\ref{e:small_derivatives}) depends on the quantity $\\sum_{|\\nu| \\leq s_0} {\\mathbf E} \\left[  | \\partial^{\\nu} F(0) |  \\right]$, thanks to our Gaussian process being stationary. For $F= \\tilde{V}_{x,T} \\mathfrak{g}_{\\alpha}$,\nLemma \\ref{l:derivative_bounds} is not directly applicable, since $F$ is no longer stationary. However, in what follows, we argue that the statement\n\\eqref{e:small_derivatives} of Lemma \\ref{l:derivative_bounds} does hold with $F$, possibly increasing the $C_{0}$ in \\eqref{e:small_derivatives}.\nFirst, the derivatives of $F$\ncould be expressed in terms of the derivatives of $\\mathfrak{g}_{\\alpha}$ and the derivatives of $\\tilde{V}_{x,T}$.\nAlso, the vector field $\\tilde{V}_{x,T}$ is asymptotically constant, as can be seen from Taylor expanding the coefficients $(a_1)_{x,T}$ and $(a_2)_{x,T}$ as in \\eqref{eq:tild(V) def},\nand using $$D(\\exp_x)_{Z = \\frac{Y}{T}} = Id + \\mathcal{O} \\left( \\frac{R}{T} \\right)$$ for $|Y|\/T \\leq R\/T$.\nHence it follows that \\eqref{e:small_derivatives} holds, with the constant $C_0$ in the r.h.s. of \\eqref{e:small_derivatives} now depending on the derivatives of these coefficients of $V$, in Riemannian normal coordinates, evaluated at $0$.\n\n\\vspace{2mm}\n\nIn preparation for the proof of Proposition \\ref{p:abc_stability}, we need an estimate on some volume quantities.\n\n\\begin{lem}[Volumes of neighborhoods of sub-$\\beta$ tangencies] \\label{l:det_cov}\nLet $x \\in \\M \\setminus \\{V=0\\}$, $A, B >0$ be small and $u \\in B(R)$ be fixed.\nThen there exists $C, \\sigma_1, \\sigma_2>0$ and $T_0(V,x)>0$ such that for all $T \\geq T_0$,\n\\begin{align*}\n& \\mathbb{P} \\left[ \\mathfrak{g}_{\\alpha} : |\\mathfrak{G}(u)| < A, | \\det D \\mathfrak{G}(u)| < B \\} \\right] \\leq \\\\\n& C A^2 \\cdot \\sqrt{\\mathcal{O}\\left(\\frac{R}{T}\\right) A^{1 - \\sigma_1} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-2\\sigma_1} + A^{1 - \\sigma_2} + B },\n\\end{align*}\nwith $\\mathfrak{G}(\\cdot)$ as in \\eqref{eq:frakG def} (implicitly depending on $x$ and $T$),\nand constants involved in the $``\\mathcal{O}\"$-notation depending on $\\alpha$ and $V$ only.\n\\end{lem}\n\\begin{proof}\nConsider the smooth Gaussian field\n\\begin{equation*}\nH:=\\left( \\mathfrak{g}_{\\alpha}, \\nabla \\mathfrak{g}_{\\alpha}, \\partial_{1,2} \\mathfrak{g}_{\\alpha}, \\partial_{1,1} \\mathfrak{g}_{\\alpha} \\right).\n\\end{equation*}\nThe distribution of $H(u)$ is non-degenerate Gaussian for every $u\\in{\\mathbb R}^{2}$ and $\\alpha\\in [0,1]$ (for $\\alpha<1$ it follows from the axiom\n$(\\rho 4^{*})$, whereas for the most subtle $\\alpha=1$ the non-degeneracy of the distribution was shown in \\cite[Appendix 1]{CMW16} via an explicit computation; it is no longer non-degenerate if $\\partial_{2,2}\\mathfrak{g}_{\\alpha}$ is added to the vector). The field $H$ proves useful for the following reasons. By rotating the plane ${\\mathbb R}^{2} \\simeq T_x \\M$\nif necessary (and using the rotation invariance of the law of $\\mathfrak{g}_{\\alpha}(\\cdot) $), we may assume that\n\\begin{equation*}\n\\tilde{V}_{x,T} = (1 + a_1(u)) \\partial_1 + a_2(u) \\partial_2\n\\end{equation*}\nwhere $a_1=a_2= \\mathcal{O}(R\/T)$ and have the additional property of $a_1(0)=a_2(0)=0$. Thus, thanks to the stationarity of $\\mathfrak{g}_{\\alpha}$ (and hence of $\\mathfrak{G}$), we have\n\\begin{align*}\n& \\mathbb{P} \\left[ \\{ |\\mathfrak{G}(u)| < A, | \\det D \\mathfrak{G}(u)| < B \\} \\right] =\n\\mathbb{P} \\left[ \\{ |\\mathfrak{G}(0)| < A, | \\det D \\mathfrak{G}(0)| < B \\} \\right] \\\\&=\n \\mathbb{P} \\left[ \\{ |(\\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0))| < A, |\\det D \\mathfrak{G}(0)| < B \\} \\right]\n\\end{align*}\nwhere\n\\begin{align*}\n\\det D \\mathfrak{G}(0) & = \\partial_2 \\mathfrak{g}_{\\alpha} \\cdot \\partial_{1,1} \\mathfrak{g}_{\\alpha} - \\partial_{1} \\mathfrak{g}_{\\alpha} \\cdot \\partial_{2,1} \\mathfrak{g}_{\\alpha} + \\left( \\partial_1 a_1 - \\partial_2 a_2 \\right) \\cdot (\\partial_1 \\mathfrak{g}_{\\alpha}) \\cdot (\\partial_2 \\mathfrak{g}_{\\alpha}) \\\\ & + \\partial_1 a_2 \\cdot (\\partial_2 \\mathfrak{g}_{\\alpha})^2 - \\partial_2 a_1 \\cdot (\\partial_1 \\mathfrak{g}_{\\alpha})^2_{|u=0}.\n\\end{align*}\n\nIt follows that,\nfor some absolute constant $c>0$,\n\\begin{align} \\label{e:ub_vol_bound}\n\\nonumber & \\mathbb{P} \\left[ \\{  |\\mathfrak{G}(u)| < A, | \\det D \\mathfrak{G}(u)| < B \\} \\right]\n \\leq cA^2 \\cdot \\mathbb{P} \\left[ \\{ | \\det D \\mathfrak{G}(0)| < B \\, \\bigg| \\, \\left| \\left( \\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0) \\right) \\right| < A  \\} \\right] \\\\\n& \\leq cA^2 \\cdot \\mathbb{P} \\left[ \\{ | \\partial_2 \\mathfrak{g}_{\\alpha} \\cdot \\partial_{1,1} \\mathfrak{g}_{\\alpha} - \\partial_{1} \\mathfrak{g}_{\\alpha} \\cdot \\partial_{2,1} \\mathfrak{g}_{\\alpha} | < B + F(u) \\, \\bigg| \\, \\left| \\left( \\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0) \\right) \\right| < A  \\} \\right],\n\\end{align}\nwhere $F(u) = \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) \\cdot A \\cdot \\partial_2 \\mathfrak{g}_{\\alpha} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) \\cdot (\\partial_2 \\mathfrak{g}_{\\alpha})^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) \\cdot A^2$ is an estimate on the remaining terms appearing the expression for $\\det D \\mathfrak{G}(u)$. Note that given $\\sigma_1, \\sigma_2 > 0$, there exist constants $C_1,C_2,c_1,c_2 >0$, independent of $A$, such that $\\mathbb{P} \\left[ \\bigg| \\partial_2 \\mathfrak{g}_{\\alpha}(0)| > A^{-\\sigma_1} \\bigg| |\\mathfrak{G}(0)| < A \\right] \\leq C_1 e^{-\\frac{c_1}{A^{2\\sigma_1}}}$ and $\\mathbb{P} \\left[ | \\partial_{2,1} \\mathfrak{g}_{\\alpha}(0)| > A^{-\\sigma_2} \\bigg| |\\mathfrak{G}(0)| < A\\right] \\leq C_2 e^{-\\frac{c_2}{A^{2\\sigma_2}}}$ thanks to the non-degeneracy of our conditional covariance matrices which follows directly from the non-degeneracy of that of $\\tilde{\\mathfrak{G}}$.\n\nLet us combine the right-hand side of \\eqref{e:ub_vol_bound} with our estimate on $F(u)$, the estimates for the probabilities $\\mathbb{P} \\left[ \\bigg| \\partial_2 \\mathfrak{g}_{\\alpha}(0)| > A^{-\\sigma_1} \\bigg| |\\mathfrak{G}(0)| < A \\right]$ and $\\mathbb{P} \\left[ | \\partial_{2,1} \\mathfrak{g}_{\\alpha}(0)| > A^{-\\sigma_2} \\bigg| |\\mathfrak{G}(0)| < A\\right]$, and the explicit quantity of $\\mathbb{P} \\bigg[ \\left| \\left( \\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0) \\right) \\right| < A \\bigg]$.  Thus we have reduced to calculating\n\\begin{align*}\n\\mathbb{P} \\bigg[ | \\partial_2 \\mathfrak{g}_{\\alpha} \\cdot \\partial_{1,1} \\mathfrak{g}_{\\alpha} | & \\leq \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) A^{1 - \\sigma_1} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-2\\sigma_1} + A^{1 - \\sigma_2} + B  \\\\\n& \\bigg| \\, \\left| \\left( \\mathfrak{g}_{\\alpha}(0), \\partial_1 \\mathfrak{g}_{\\alpha}(0) \\right) \\right| < A \\bigg].\n\\end{align*}\nThe fact that at least one of the terms $\\partial_2 \\mathfrak{g}_{\\alpha}, \\partial_{1,1} \\mathfrak{g}_{\\alpha}$ must be bounded above by the squareroot of the right-hand side along with the non-degeneracy of the corresponding conditional covariance matrix gives the desired upperbound of\n\\begin{equation*}\n2 \\sqrt{\\mathcal{O}\\left(\\frac{R^2}{T^2}\\right) A^{1 - \\sigma_1} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-2\\sigma_1} + A^{1 - \\sigma_2} + B }.\n\\end{equation*}\nNow, set $\\sigma_1=\\sigma_2=1\/2$ (although this may not be an optimal choice of parameters).\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{p:abc_stability}]\n\nOur proof follows the proof of \\cite[Proposition 4.7]{SW18} with some important modifications of the argument. The main idea is to show that if the count for critical points of $\\mathfrak{G}$, that are also low-lying in the sense that the first component of $\\mathfrak{G}$ is small, is too large, then for sufficiently small parameters $\\beta_{2,2}$, the smoothness of the field $\\mathfrak{G}$ allows us to derive contradictions in volume comparisons for neighborhoods of such critical points.\n\nFirst, we restrict ourselves to the event $\\left( \\cap_{i=1}^6 \\Delta_i^{\\complement}\\right)$ where $R \\geq R_0(\\eta,\\delta)$, $\\beta_{1,3}=\\beta_{1,3}(\\eta,\\delta)$; the parameters governing the events $\\Delta_i$, $i=1,...,5$, have been chosen such that $\\cap_{i=1}^5 \\Delta_i^{\\complement}$ has probability $> 1 - \\frac{\\delta}{8}$. Note that $\\mathbb{P}[\\Delta_6(\\eta, R, \\beta_{1,3})^{\\complement}] > 1 - \\frac{\\delta}{8}$. We are now in the situation where the nodal components of $\\mathfrak{g}_{\\alpha}$ are stable in the sense of Definition \\ref{d:unif_stability_fields}.\nFor a parameter $\\beta_{2,2}$, we let $K$ be the number of $\\beta_{2,2}$-unstable balls for $\\mathfrak{G}$. We will eventually be able to find a number $\\beta_{2,2}$ that satisfies the conclusion of Proposition \\ref{p:unstable_tangencies}.\n\nNext, we introduce a small parameter $\\gamma = \\gamma(R_1)$ that we will specify towards the end of our proof. We set\n\\begin{equation}\n\\label{eq:Ktild=K}\n\\tilde{K} = c_1 K\n\\end{equation}\nafter invoking the separation statement Lemma \\ref{l:4_separation}, and consider the new associated covering $\\{ \\mathcal{G}_{i_j} \\}_{i_j \\leq \\tilde{K}}$, with centers $u_{i_j}$, where we have possibly reordered the indices $i_j \\leq \\tilde{K}$. Now Taylor expand $\\det D \\mathfrak{G}$ to obtain the bound\n\\begin{equation} \\label{e:DG_expansion}\n|\\det D \\mathfrak{G}(u)| < \\beta_{2,2} + C_2 \\bigg( \\sup\\limits_{|\\nu| = 1, |\\tau|=1 ,\\, x \\in B(u_{i_j}, \\gamma)} |\\partial^{\\nu}  \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha}(u) | \\, |\\partial^{\\tau}  \\tilde{V}_{x,T}\\mathfrak{g}_{\\alpha}(u)| \\bigg) \\cdot \\gamma\n\\end{equation}\non each ball $B(u_{i_j}, \\gamma)$.\nAs a result of an application of Lemma \\ref{l:derivative_bounds} on \\eqref{e:DG_expansion}, we obtain the numbers\n\\begin{align*}\nA &:=  \\beta_{2,2} + c_2 \\frac{R}{\\tilde{\\sqrt{K}}}\\cdot \\gamma \\\\\nB &:= \\beta_{2,2} + c_3^2 \\frac{R^2}{\\tilde{K}} \\gamma\n\\end{align*}\nwith $c_{2}, c_3$ absolute, so that with probability $1 - \\delta\/4$ for $R \\geq R_1$, the bounds $|\\mathfrak{G}(u)| < A$ and $|\\det D \\mathfrak{G}(u)| < B$ hold on at least half of the $B(u_j, \\gamma)$, assuming $R \\geq R_1$\n(though we may neglect the distinction between ``all of the balls\" and ``half of them\" by appropriately modifying\nthe constant in \\eqref{eq:Ktild=K}).\n\n\nOur third, and most important, step is to manipulate the random variable\n\\begin{equation*}\n\\mathcal{A}_{A,B}  = \\mbox{Area} \\left( \\{ u:  |\\mathfrak{G}(u)|<A, |\\det D\\mathfrak{G}(u)| < B \\} \\right)\n\\end{equation*}\nin order to obtain a sufficient estimate on $K$ and deduce some necessary restrictions on $\\beta_{2,2}$.\nWe now have with probability $> 1 - \\delta\/4$, following immediately from the assumptions on the balls $B(u_j, \\gamma)$ and that $\\tilde{K} = c_1K$, that there exists $c_3 > 0$ uniform such that the following bound holds:\n\\begin{equation}\n\\label{eq:Ac(A,B)>=cgamma^2}\n\\mathcal{A}_{A,B} \\geq c_3 \\gamma^2 \\cdot K.\n\\end{equation}\n\nWe have\n\\begin{equation*}\n\\mathcal{A}_{A,B} = \\int\\limits_{B(R)} \\chi_{A,B} \\left( g_1,g_2, \\frac{\\partial g_1}{\\partial x_1}  \\frac{\\partial g_2}{\\partial x_2} -  \\frac{\\partial g_1}{\\partial x_2} \\frac{\\partial g_2}{\\partial x_1} \\right) \\, dx\n\\end{equation*}\nwhere $\\chi_{A,B}(\\cdot, \\cdot, \\cdot)$ is the indicator function for the set $\\{|(v_1, v_2)| < A, \\, |v_3| < B \\}$. If we can obtain a sufficient bound on ${\\mathbf E}[\\mathcal{A}_{A,B}]$ then, after an application of Chebyshev's Inequality, we have a sufficient estimate on $\\mathcal{A}_{A,B}$ with high probability. This is where Lemma \\ref{l:det_cov} is invoked, since\n\\begin{equation*}\n{\\mathbf E} \\left[\\chi_{A,B} \\left( g_1,g_2, \\frac{\\partial g_1}{\\partial x_1}  \\frac{\\partial g_2}{\\partial x_2} -  \\frac{\\partial g_1}{\\partial x_2} \\frac{\\partial g_2}{\\partial x_1} \\right) \\right] =  \\mathbb{P} \\left[ \\mathfrak{g}_{\\alpha} : |\\mathfrak{G}(u)| < A, | \\det D \\mathfrak{G}(u)| < B \\} \\right].\n\\end{equation*}\n\nWe remind ourselves (as already done in the proof of Theorem \\ref{t:conv_prob}) that in the local coordinates around $x$ given by $y=\\exp_x(Y)$, which allows us to identify ${\\mathbb R}^2$ with $T_xM$, we set $\\tilde{V}(x)$ to be the constant vector field given by the trivial extension of $(\\tilde{V}_{x,T})_{|Y=0}$ to ${\\mathbb R}^2$.  We can choose $T \\geq T_1(V(x))$ such that $\\tilde{V}_{x,T} = \\tilde{V}(x) + o(1)$; we remind ourselves that we have also assumed $\\tilde{V}(x) \\neq 0$. Note our $T_1$ can be taken to be greater than the $T_0$ appearing in the statement of Lemma \\ref{l:det_cov}. Let $s_{1},s_{2}$ be any positive numbers satisfying $2s_{1}+s_{2}=2$, so that $\\gamma^{-2} = \\gamma^{-2s_1 - s_2}$.\nThe estimate given by Lemma \\ref{l:det_cov} together with \\eqref{eq:Ac(A,B)>=cgamma^2}, yields that\n\\begin{align} \\label{e:penult_step_small_number_K}\n\\nonumber K & \\leq C_3 \\gamma^{-2}  \\,  \\left(  \\beta_{2,2} + c_2 \\frac{R}{\\sqrt{\\tilde{K}}} \\gamma  \\right)^2 \\\\\n\\nonumber & \\times \\sqrt{\\mathcal{O}\\left(\\frac{R}{T}\\right) A^{1\/2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2 + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-1} + A^{1\/2} +  \\left( \\beta_{2,2} + c_3^2 \\frac{R^2}{\\tilde{K}} \\gamma \\right)  } \\times R^2 \\\\\n\\nonumber & = C_3 \\,  \\left(\\beta_{2,2}\\gamma^{-s_1} + c_2 \\frac{R}{\\sqrt{\\tilde{K}}} \\gamma^{1-s_1}   \\right)^2 \\cdot \\bigg( \\mathcal{O}\\left(\\frac{R}{T}\\right) A^{1\/2} \\gamma^{-2s_2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2\\gamma^{-2s_2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-1} \\gamma^{-2s_2} \\\\\n& + A^{1\/2}  \\gamma^{-2s_2}+  \\left( \\beta_{2,2}\\gamma^{-2s_2} + c_3^2 \\frac{R^2}{\\tilde{K}} \\gamma^{1-2s_2} \\right)  \\bigg)^{\\frac{1}{2}} \\times R^2.\n\\end{align}\nIn preparation for the final steps of our proof, let us define\n\\begin{align} \\label{e:xi_value}\n\\nonumber & \\xi := C_3 \\left(\\beta_{2,2}\\gamma^{-s_1} + c_2 \\frac{R}{\\sqrt{\\tilde{K}}} \\gamma^{1-s_1}   \\right)^2 \\cdot \\bigg( \\mathcal{O}\\left(\\frac{R}{T}\\right) A^{1\/2} \\gamma^{-2s_2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^2\\gamma^{-2s_2} + \\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-1} \\gamma^{-2s_2} \\\\\n& + A^{1\/2}  \\gamma^{-2s_2}+  \\left( \\beta_{2,2}\\gamma^{-2s_2} + c_3^2 \\frac{R^2}{\\tilde{K}} \\gamma^{1-2s_2} \\right)  \\bigg)^{\\frac{1}{2}}\n\\end{align}\nso that we have $K \\leq \\xi R^2$; without loss of generality, if necessary, we increase $C_3$ to be greater than $1$.\n\nLet us now proceed by contradiction. To wit, let us assume that $K > \\eta R^2$ from which it follows that $\\frac{R}{\\sqrt{\\tilde{K}}} < \\frac{1}{c_1 \\sqrt{\\eta}}$, on recalling \\eqref{eq:Ktild=K}. In order to make our choice of parameters $\\gamma, \\beta_{2,2},$ and $T_0$ properly so that we can eventually make $\\xi < \\eta$, we proceed as follows.\n\\begin{enumerate}\n\\item To start, we let $\\gamma <1$ and $\\beta_{2,2} < 1$.\nWe would like to make $A^{1\/2} \\gamma^{-2s_2} < \\frac{1}{100}$. Therefore when given $\\eta$, this leads us the first restriction of\n\\begin{equation*}\n\\gamma^{1-4s_2} < \\frac{1}{100} \\frac{c_1 \\sqrt{\\eta}}{c_2},\n\\end{equation*}\ntherefore requiring that $1 - 4s_2 > 0 $.\n\\item We aim to have that $A^2 \\gamma^{-2s_1} =  \\left(\\beta_{2,2}\\gamma^{-s_1} + c_2 \\frac{R}{\\sqrt{\\tilde{K}}} \\gamma^{1-s_1}   \\right)^2 $ in the r.h.s of (\\ref{e:xi_value}) is less than $\\frac{\\eta}{100C_3}$. This requires that\n\\begin{equation*}\n\\gamma^{1-s_1} < \\frac{c_1 \\,\\eta}{100 \\, c_2 \\, C_3}\n\\end{equation*}\ntherefore requiring that $1 - s_1 > 0 $. As we have the initial restriction of $2s_1 + s_2 =2$, as a concrete working set of parameters, we make the choice of $s_1 = 15\/16$ and $s_2 = 1\/8$. Hence, we take\n\\begin{equation*}\n\\gamma  < \\min \\left\\{ \\left(\\frac{c_1 \\, \\sqrt{\\eta}}{100 \\, c_2 \\, C_3} \\right)^{2}, \\left(\\frac{c_1 \\eta}{100 \\, c_2 \\, C_3} \\right)^{16} \\right\\} = \\left( \\frac{c_1}{100 c_2 C_3} \\right)^{16} \\eta^{16}.\n\\end{equation*}\n\\item With the goals of eventually having $A^2 \\gamma^{-2s_1} < \\frac{\\eta}{100C_3}$ in the r.h.s of (\\ref{e:xi_value}) and the second (non-constant) factor that is raised to the power of $1\/2$ be $< 1$, we make another restriction of\n\\begin{align*}\n& \\beta_{2,2} < \\min \\left\\{ \\frac{\\eta^{1\/2}}{\\sqrt{100C_3}} \\gamma^{s_1}, \\frac{1}{10} \\gamma^{2s_2} \\right\\} =  \\min \\left\\{ \\frac{\\eta^{1\/2}}{100C_3} \\gamma^{15\/16}, \\frac{1}{100} \\gamma^{1\/4} \\right\\} \\\\\n& = \\frac{\\eta^{1\/2}}{100C_3} \\gamma^{15\/16} < \\frac{1}{100C_3} \\,  \\left( \\frac{c_1}{100 c_2 C_3} \\right)^{15} \\eta^{31\/2}.\n\\end{align*}\nafter using the restriction on $\\gamma$.  Note that our restrictions on $\\gamma$ and $\\beta$ make all terms which do not depend on $T$ sufficiently small.\n\\item Now, considering all the terms in (\\ref{e:xi_value}) involving factors of $T^{-1}$, appropriately bounding the term $\\mathcal{O}\\left(\\frac{R^2}{T^2}\\right)A^{-1} \\gamma^{-2s_2}$ above by $\\frac{1}{100}$ will place similar bounds on the aforementioned remaining terms.  Letting $C$ be the supremum amongst all the implicit constants depending on a derivative of $V$ (that is, the $C^1$ norm of $V$'s coefficients) and appearing in the $\\mathcal{O}$ notations, we make the final restriction of\n\\begin{equation*}\n\\frac{C}{T} < \\frac{\\sqrt{A}}{10R} \\gamma^{2s_2}= \\frac{\\sqrt{A}}{10R} \\gamma^{1\/4} < \\frac{1}{10R} \\left(\\frac{c_1}{c_2 C_3} \\right)^4 \\eta^{15\/4}.\n\\end{equation*}\n\\end{enumerate}\nHence, we enlarge $T_0$ to satisfy our final constraint (and therefore inheriting a possible dependence on $R$). With all these choices made, in the order described above, we have arrive at $\\xi < \\frac{\\eta}{2500}$ and therefore contradict the hypothesis of $K > \\eta R^2$. This concludes the proof of Proposition \\ref{p:abc_stability}.\n\\end{proof}\n\nWe now collect some facts on nodal components that fall outside of our local approximation and are exceptional in terms of the volumes of the nodal domains for which they form the boundary. But first, we record some necessary definitions:\n\n\\begin{defn}\n\\begin{enumerate}\n\\item We say that a nodal component of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ is $\\xi$-small if it is adjacent to a domain of volume $< \\xi$.\n\n\\item For $R >0$, let $\\pazocal{N}_{\\xi-sm}(\\mathfrak{g}_{\\alpha}, R)$ be the number of $\\xi$-small components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ lying entirely inside of $B(R)$.\n\n\\item We say that a nodal component of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ is $R_1$-long if its diameter is $> R_1$.\n\n\\item For $R > R_1 >0$, let $\\pazocal{N}_{diam > R_1}(\\mathfrak{g}_{\\alpha};R)$ be the number of $R_1$-long components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ lying entirely inside of $B(R)$.\n\\end{enumerate}\n\\end{defn}\n\nThe following results are taken directly from \\cite{So12}:\n\n\\begin{lem}[Cf. {\\cite[Lemma 8]{So12}}] \\label{l:long_components}\nConsider $\\mathfrak{g}_{\\alpha}$ on $B(R)$. Let $\\delta>0$ be given. Then given $R_1 >0$, there exists $R_0(R_1)$ such that for all $R \\geq R_0 \\geq R_1$ and a uniform constant $C>0$, such that the number of components $\\Gamma \\subset B(R)$ of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ of diameter $> R_1$ is\n\\begin{equation*}\n\\pazocal{N}_{diam > R_1}(\\mathfrak{g}_{\\alpha};R) < \\frac{C}{R_1} R^2\n\\end{equation*}\nwith probability $> 1 - \\delta$.\n\\end{lem}\n\nIn our $2$-dimensional case Lemma \\ref{l:long_components} follows from the isoperimetric inequality; in the more general setting a slightly heavier\nmachinery involving the restriction of $\\mathfrak{g}_{\\alpha}$ to the boundary of $B(R)$ is employed. We note that it, thanks to discarding the long components, coverings of $B(R)$ with balls of radius $R_1 < R$ as in the previous sequence of statements will give us an advantage.\n\n\n\\begin{lem}[Cf. {\\cite[Lemma 9]{So12}}] \\label{l:small_vol}\nLet $\\xi>0$ be given.  There exist constants $c_0, C_0>0$ such that\n\\begin{equation*}\n\\limsup\\limits_{R \\rightarrow \\infty} \\frac{{\\mathbf E} \\left[ \\pazocal{N}_{\\xi-sm}(\\mathfrak{g}_{ \\alpha}, R) \\right]}{R^2} \\leq C_0 \\, \\xi^{c_0}.\n\\end{equation*}\n\\end{lem}\n\nGiven our facts about $\\xi$-small and $R_1$-long components, we can now state the following important lemma:\n\n\\begin{lem} \\label{l:sub_tang_low_count}\nLet $x \\in \\M\\setminus\\{V=0\\}$. Given the parameters $R, \\delta, \\eta_0>0$, there exists a covering radius parameter $R_{1,0} < R$, an auxiliary density parameter $\\eta_1(R, \\eta_0) < \\eta_0$, an event $E$ with $\\mathbb{P}[E] < \\delta$ and a stability parameter $\\beta_{2,2}$ such that outside $E$, if $F$ is $(\\eta_1, \\beta_{2,2}, 3R_{1,0})$-stable for all $T \\geq T_0(R)$, then $|\\det D \\mathfrak{G}| > \\beta_{2,2}$ on all but $\\eta_0 R^2$ components of $\\mathfrak{g}_{\\alpha}^{-1}(0)$ that are also uniformly stable as in Definition \\ref{d:unif_stability_fields}.\n\\end{lem}\n\n\\begin{proof}\nOur proof is almost verbatim that of \\cite[Lemma 4.8]{SW18} which is similar in spirit to that in \\cite[Section 4.2]{NS09}. We refer the interested reader to this paper but highlight the main details as well as an explicit formula which gives further insight into the dependences between our variety of parameters. To ease the notation and intuitively connect the parameters in this lemma with those in our previous statements, we replace $R_{1,0}$ by $R_1$ in our proof.\n\nThanks to lemmas \\ref{l:long_components} and \\ref{l:small_vol}, we know that with probability $1  - \\frac{\\delta}{2}$ there exists uniform constants $C_1,C_2, c_0>0$ such that we have that\n\\begin{equation} \\label{e:long_toosmall_comps}\n\\pazocal{N}_{diam>R_1}(\\mathfrak{g}_{\\alpha}, R) \\leq C_1 \\, \\frac{R^2}{R_1} \\, \\mbox{ and } \\, \\pazocal{N}_{\\xi_0-sm}(\\mathfrak{g}_{\\alpha}, R) \\leq C_2 \\, \\xi^{c_0} \\, R^2\n\\end{equation}\nfor some $R_1 < R$, where $R \\geq R_0$. This reduces us to now having to count components that are not $R_1$-long or are not $\\xi_0$-small.  This particular $R_1$ is what we will take as our covering radius parameter, for at least the time being.\n\nProposition \\ref{p:unstable_comp} states that with probability $1 - \\frac{\\delta}{2}$ and $R_0$ possibly larger, the total number of components that are not long or small in the above senses and are stable in terms of having low-lying zeroes of $\\nabla \\mathfrak{g}_{\\alpha}$ is $\\leq \\eta_1 R^2$.  Now, we apply Proposition \\ref{p:abc_stability} and the uniform lower bound we have on the volume encompassed by the individual components to see that the number of components that are neither long nor small with no sub-$\\beta_{2,2}$ $\\tilde{V}_{x,T}$- tangencies is $\\leq C_3 \\frac{R_1^2}{\\xi_0} \\frac{\\eta_1}{2} R^2$ for some uniform $C_3 >0$. Summing up all these counts gives an upper bound for the number of components that are either unstable in the sense of Definitions \\ref{d:unif_stability_fields} and \\ref{d:unstable_event} and those that are not amenable to our local methods.\n\nBy summing all of our declared estimates appearing in (\\ref{e:long_toosmall_comps}) and the previous paragraph, and therefore collating all the parameters we can adjust, we aim to establish the following inequality:\n\\begin{equation*}\nC \\left( \\frac{1}{R_1} + \\xi_0^{c_1} + \\frac{R_1^2}{2 \\xi_0}\\eta_{1}  + \\eta_1 \\right) < \\eta_0\n\\end{equation*}\nfor some uniform constants $C, c_1>0$. Motivated by this goal, we can choose the range for our medley of parameters in the following order: our radius parameter $R_1 > \\frac{4C}{ \\eta_0}$, $\\xi_0 < (\\frac{\\eta_0}{4C})^{1\/c_1}$, and finally our auxiliary density parameter $\\eta_1 < \\min \\{\\frac{\\eta_0}{4C}, \\frac{2 \\xi_0 }{R_1^2} \\frac{\\eta_0}{4C} \\}$. Our choice of $T_0$ comes from Proposition \\ref{p:abc_stability}. This gives our desired result.\n\\end{proof}\n\n\\begin{rem}\nNote that in the above proof, our choice of parameters $R_1 = R_{1,0}, \\xi_0, \\eta_1$ is not dictated by our choice of stability parameter $\\beta_{2,2}$ and hence not making our reductions circular. However, these choices of parameters could possibly make $\\beta_{2.2}$ smaller. Given the restrictions we derived in the proof of Proposition \\ref{p:abc_stability}, this does not harm our overall argument.\n\\end{rem}\n\nFinally, we can deduce Proposition \\ref{p:unstable_tangencies} from Lemma \\ref{l:sub_tang_low_count}.\n\n\\begin{proof}[Proof of Proposition \\ref{p:unstable_tangencies}]\nWe apply Lemma \\ref{l:sub_tang_low_count} directly. Considering that we are given $\\eta, \\delta>0$ and $x$, we take any $R_0 > R_{1,0}$ such that $\\frac{R_0}{R_{1,0}}$ is large, take $\\eta_0 < \\eta$, set $\\beta_2 = \\beta_{2,2}$, and take the same $T_0$ as provided.\n\\end{proof}\n\nWe close this section by emphasizing that our count of $\\eta R^2$ is for the total number of components with at least one near-degenerate tangency.  Thus, there is still room to make more precise this upper bound for those with precisely $k$ tangencies with at least one being near-degenerate.\n\n\n\n\\section{Proof of Theorem \\ref{thm:main meas conv}: global results in the Riemannnian scenario}\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:main meas conv} assuming Theorem \\ref{t:main_thm_less_equal_1}: consolidating the individual counts:}\n\n\\label{sec:proof main thm meas}\n\n\\begin{proof}\n\nUsing the constants $C_{\\alpha,k}$ prescribed by Theorem \\ref{t:main_thm_less_equal_1},\nwe define the measure\n\\begin{equation}\n\\label{eq:mualpha def}\n\\mu_{\\alpha}:=\\sum\\limits_{k=0}^{\\infty}C_{\\alpha,k}\\cdot\\delta_{k},\n\\end{equation}\nand Theorem \\ref{t:main_thm_less_equal_1} part (1) reads for every $k\\ge 0$,\n\\begin{equation*}\n{\\mathbf E}[|\\left(\\mu_{f_{\\alpha,T}}(V)\\right)(k) - \\mu_{\\alpha}(k) |] \\rightarrow 0\n\\end{equation*}\nas $T\\rightarrow\\infty$.\nWe claim that Theorem \\ref{thm:main meas conv} holds true with $\\mu_{\\alpha}$ as in \\eqref{eq:mualpha def}.\nFirst, since by the proof of Theorem \\ref{t:main_thm_less_equal_1} below, the constants $C_{\\alpha}$ are\nsame as in Theorem \\ref{thm:euclid_count} (1)\napplied on the random field $F=\\mathfrak{g}_{\\alpha}$ with $\\zeta$ arbitrary (cf. Proposition \\ref{p:almost_main_theorem}), where\nthe $C_{k}$ are independent of $\\zeta$ by Theorem \\ref{thm:euclid_count} (2), the measure\n$\\mu_{\\alpha}$, as defined in \\eqref{eq:mualpha def}, is a probability measure, thanks to Theorem \\ref{thm:euclid_count} (4).\n\nTherefore $\\{\\mu_{f_{\\alpha,T}}(V)\\}_{T}$ on ${\\mathbb Z}_{\\ge 0}$ is a collection of random probability measures,\nand $\\mu_{\\alpha}$ a deterministic probability measure, so that,\nfor every $k\\ge 0$, as $T\\rightarrow\\infty$, $(\\mu_{f_{\\alpha,T}}(V))(k)$ converges in mean (and hence in probability) to $\\mu_{\\alpha}(k)$.\nThis is precisely the scenario considered by ~\\cite[Lemma 6.3]{ALL}, proof (and same result)\nalready contained within ~\\cite[Proof of Theorem 1.1, \\S 7.1 on p. 57]{SW18},\nwhose conclusion yields the convergence in probability of\n$\\{\\mu_{f_{\\alpha,T}}(V)\\}_{T}$ as $T\\rightarrow\\infty$ to $\\mu_{\\alpha}$ w.r.t. the total variation distance \\eqref{eq:D tot var dist def},\ni.e. it gives \\eqref{eq:main meas conv}.\nFinally, the support statement of Theorem \\ref{thm:main meas conv} for $\\mu_{\\alpha}$ is an immediate corollary from\nthe definition \\eqref{eq:mualpha def} of $\\mu_{\\alpha}$ and Theorem \\ref{t:main_thm_less_equal_1} part (2).\nTheorem \\ref{thm:main meas conv} is now proved.\n\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{t:main_thm_less_equal_1}}\n\n\\begin{defn}\\label{def:plus_minus}\nFor $y \\in {\\mathbb R}$, we denote\n\\begin{equation*}\n|y|_+ = \\max \\{0,y\\}\n\\end{equation*}\nand\n\\begin{equation*}\n|y|_-= \\min \\{0,y\\},\n\\end{equation*}\nso that\n\\begin{equation*}\n| \\cdot | = | \\cdot |_+ + | \\cdot |_-.\n\\end{equation*}\n\\end{defn}\n\nLet $V \\in \\mathcal{V}(\\M)$ throughout this section. We remind ourselves that\n\\begin{equation*}\n\\pazocal{N}_{V}(f,k) :=\\#\\{\\gamma  \\subseteq f^{-1}(0):\\: \\pazocal{N}_{V}(\\gamma)=k\\}\n\\end{equation*}\nis the total number of nodal components of $f$ with precisely $k$ tangencies w.r.t. $V$. We now present a key proposition that builds upon on our previous (and crucial) local results and essentially gives the conclusions of our main theorems:\n\n\\begin{prop} \\label{p:almost_main_theorem}\nLet $k \\ge 0$ be given and $c_{2, \\alpha}$ be the Nazarov-Sodin constant of $\\mathfrak{g}_{\\alpha}$,\nand the constants $C_{\\alpha,k}$ prescribed by an application of Theorem \\ref{t:main_thm_less_equal_1} (1) on the\nfield $\\mathfrak{g}_{\\alpha}$ (with $\\zeta$ arbitrary). Then, as $T \\rightarrow \\infty$, we have that\n\\begin{equation*}\n{\\mathbf E} \\left[  \\left| \\frac{ \\pazocal{N}_{V}\\left( f_T,  k  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(\\M) \\cdot T^2} - C_{\\alpha,k} \\right|_{\\pm}  \\right] \\rightarrow 0.\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}[Proof of Theorem \\ref{t:main_thm_less_equal_1} assuming Proposition \\ref{p:almost_main_theorem}]\n\nTheorem \\ref{t:main_thm_less_equal_1} follows immediately from Proposition \\ref{p:almost_main_theorem}.\n\n\\end{proof}\n\n\\subsection{Excising small volume and long components}\n\n\\begin{defn}\nLet $\\xi, D> 0$ be parameters and $f_T$ the band-limited functions \\eqref{eq:fT band lim alpha<1} (or \\eqref{eq:fT band lim alpha=1}).\n\n\\begin{enumerate}\n\\item A component of $f^{-1}_T(0)$ is \\textit{$\\xi$-small} if it is a boundary of nodal domain whose volume in $\\M$ is less than $\\xi T^{-2}$.  Let $\\pazocal{N}_{\\xi-sm}(f_T)$ be the total number of $\\xi$-small components of $f_T$ on $\\M$.\n\\item For $D>0$, a component of $f^{-1}_T(0)$ is \\textit{$D$-long} if its diameter is greater than $D\/T$. Let $\\pazocal{N}_{D-long}(f_T)$ be their total number.\n\\item Given the parameters $D, \\xi>0$, a component of $f^{-1}_T(0)$ is \\textit{$(D, \\xi)$-normal}, if it is neither $\\xi$-small nor $D$-long.\n\\item Let $\\pazocal{N}_{norm}(f_T)$ be the total number of $(D,\\xi)$-normal components of $f^{-1}_T(0)$ (analogously, define $\\pazocal{N}_{V, norm}(f_T, k)$ as the count for the subset of $(D,\\xi)$-normal components whose number of tangencies is precisely $k$ with respect to $V$).\n\\item  Let $\\pazocal{N}_{V, norm}(f_T, x, r)$ be the total number of $(D,\\xi)$-normal components of $f^{-1}_T(0)$ that are completely contained inside the geodesic ball $B(x;r)$ (respectively, $\\pazocal{N}_{V,norm}(f_T, k, x, r)).$\n\\item  Let $\\pazocal{N}^*_{norm}(f_T, x, r)$ be the total number of $(D,\\xi)$-normal components of $f^{-1}_T(0)$ that intersect the geodesic ball $B(x;r)$ (respectively, $\\pazocal{N}^*_{V,norm}(f_T, k, x, r)).$\n\\end{enumerate}\n\\end{defn}\n\nThe following two lemmas are taken directly out of \\cite{So12}:\n\n\\begin{lem}[Cf. {\\cite[Lemma 9]{So12}}]\nThere exists constants $c_0, C_0 >0$ so that the following estimate on the number of $\\xi$-small components holds:\n\\begin{equation*}\n\\limsup\\limits_{T \\rightarrow \\infty} \\frac{{\\mathbf E} \\left[  \\pazocal{N}_{\\xi-sm}(f_T) \\right]}{T^2} \\leq C_0 \\, \\xi^{c_0}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{lem}[Cf. {\\cite[Lemma 8]{So12}}]\nThere exists a constant $C_0 >0$ such that the following bound holds on the number of $D$-long components:\n\\begin{equation*}\n\\limsup\\limits_{T \\rightarrow \\infty} \\frac{{\\mathbf E} \\left[  \\pazocal{N}_{D-long}(f_T) \\right]}{T^2} \\leq C_0 \\, \\frac{1}{D}.\n\\end{equation*}\n\\end{lem}\n\n\nWe can now state the main proposition of this section:\n\n\\begin{prop} \\label{p:conv_in_mean_normal}\nLet $\\xi, D>0$ be given and $k \\ge 0$. Then as $T \\rightarrow \\infty$, we have that\n\\begin{equation*}\n{\\mathbf E} \\left[  \\left| \\frac{ \\pazocal{N}_{V,norm}\\left( f_T,  k  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(\\M) \\cdot T^2} - C_k \\right|_{\\pm}  \\right] \\rightarrow 0,\n\\end{equation*}\nfor $| \\cdot |_{\\pm}$ as in Definition \\ref{def:plus_minus}, and $C_{k}=C_{\\alpha,k}$ is in Proposition \\ref{p:almost_main_theorem}.\n\\end{prop}\n\n\\begin{proof}[Proof of Proposition \\ref{p:almost_main_theorem} assuming Proposition \\ref{p:conv_in_mean_normal}]\nThis is immediate given lemmas \\ref{l:small_vol} and \\ref{l:long_components}.\n\\end{proof}\n\n\\subsection{Proof of Proposition \\ref{p:conv_in_mean_normal}}\n\nWe begin with a global analogue of Lemma \\ref{l:integro_geom_sandwich}.\n\n\\begin{lem} \\label{l:integro_geom_sandwich_global}\nGiven $\\epsilon > 0$ and $k \\ge 0$, there exists $r_0$ such that for all $r < r_0$, we have\n\\begin{align*}\n& (1 - \\epsilon) \\int\\limits_{\\M} \\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, r  \\right)}{\\operatorname{Vol}(B(r))} \\, dx \\leq \\pazocal{N}_{V,norm}\\left( f_T,  k \\right)\\leq (1 + \\epsilon) \\int\\limits_{\\M} \\frac{\\pazocal{N}^*_{V,norm}\\left( f_T,  k, x, r  \\right)}{\\operatorname{Vol}(B(r))} \\, dx.\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}\nAs the proof is almost exactly the same as that of Lemma \\ref{l:integro_geom_sandwich}, we omit it and leave the details to the reader.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{p:conv_in_mean_normal}]\nOnce again, our proof mirrors that of \\cite[Proposition 3]{BW17} so we give the major steps of the proof, with particular details given at the critical junctures, and leave the remaining similar details to the interested reader.  For convenience, we assume that $\\operatorname{Vol}(\\M)=1$.\nLet $\\epsilon > 0$ and $k \\ge 0$ be given. Consider a regime of $R,T$ such that $R\/T < r_0 < 1$ as in Lemma \\ref{l:integro_geom_sandwich_global} and set $D = \\sqrt{R}$ so that\n\\begin{equation*}\n\\frac{\\operatorname{Vol}(B(R+D))}{\\operatorname{Vol}(B(R))} < 1 + \\epsilon.\n\\end{equation*}\n\nWe focus on the case for $| \\cdot |_+$ as the proof for $| \\cdot |_{-}$ is similar. We begin by applying Lemma \\ref{l:integro_geom_sandwich_global} with $r = R\/T$. It follows that\n\\begin{align*}\n {\\mathbf E} \\left[  \\left| \\frac{ \\pazocal{N}_{V,norm}\\left( f_T,  k  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(\\M) \\cdot T^2} - C_k \\right|_{+}  \\right] &\\leq {\\mathbf E} \\left[  \\int\\limits_{\\M} \\left|(1 + 2\\epsilon) \\frac{\\pazocal{N}^*_{V,norm}\\left( f_T,  k, x, R\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} \\, dx - C_k  \\right|_{+} \\, dx  \\right] \\\\\n& \\leq {\\mathbf E} \\left[  \\int\\limits_{\\M} \\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+} \\, dx  \\right] \\\\\n& + \\mathcal{O} \\left( \\epsilon \\cdot \\int\\limits_{\\M} \\frac{{\\mathbf E} \\left[\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right) \\right]}{ c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} \\, dx \\right).\n\\end{align*}\nNote that\n\\begin{equation*}\n\\int\\limits_{\\M} \\frac{{\\mathbf E} \\left[\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right) \\right]}{ c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} \\, dx  =  \\mathcal{O}(1)\n\\end{equation*}\nby \\cite[Lemma 2]{So12} on using Lemma \\ref{lem:Kac Rice manifold} to obtain\nan upper bound for the total number of components. Thus, our focus is on proving that\n\\begin{equation*}\n{\\mathbf E} \\left[  \\int\\limits_{\\M} \\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+} \\, dx  \\right]  \\rightarrow 0.\n\\end{equation*}\n\nLet $\\rho < \\frac{\\epsilon}{5\\sqrt{\\pi} N}$, where $N$ is the cardinality of $\\{ V = 0 \\}$, be a parameter that will control how close we are allowed to approach the zero set $\\{V=0\\}$. Let $\\xi >0$ be a small number, to be specified shortly, that controls how small a volume a nodal component can encompass and will also be related to our given $\\epsilon$ later.\nWe now consider the event\n\\begin{equation*}\n\\Delta_{T,k,x,R} := \\left\\{ \\left| \\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))}  - C_k  \\right| > \\frac{\\epsilon}{5} \\right\\}.\n\\end{equation*}\nLet $\\{ V= 0\\}^{\\sqrt{\\rho\/2}}$ be a neighborhood consisting of balls of radius $\\sqrt{\\rho\/2} < inj(\\M)$ centered at the finitely many points of $\\{V=0\\}$. An application of an Egorov-type Theorem w.r.t. the double limit $\\lim\\limits_{R\\rightarrow\\infty}\\lim\\limits_{T\\rightarrow\\infty}$\n(as carried out in \\cite[Proposition 3]{BW17}), along with Theorem \\ref{t:conv_prob} allows us to conclude that on $\\M \\setminus (\\{ V= 0\\}^{\\sqrt{\\rho\/2}})$, there exists a set $\\M_{\\epsilon} \\subset \\M$ with $\\operatorname{Vol}(\\M_{\\epsilon}) > 1 - \\sqrt{\\pi} \\frac{\\epsilon}{5}$ (with the additional property that $\\M_{\\epsilon} \\cap \\{ V= 0\\}^{\\sqrt{\\rho\/2}} = \\emptyset$ thanks to taking a tubular neighborhood of radius $\\sqrt{\\rho\/2}$), such that\n\\begin{equation}\n\\label{eq:double limit liminf=0}\n\\liminf_{R \\rightarrow \\infty} \\limsup_{T \\rightarrow \\infty} \\sup\\limits_{x \\in \\M_{\\epsilon}} \\mathbb{P}\\left[  \\Delta_{T,k,x,R} \\right] = 0,\n\\end{equation}\ni.e. the limit \\eqref{eq:double lim R,T conv loc} is {\\em almost} uniform along some sequence $R_{j}\\rightarrow\\infty$ attaining the $\\liminf\\limits_{R\\rightarrow\\infty}$ in \\eqref{eq:double limit liminf=0}, so that\n\\begin{equation}\n\\label{eq:double limit R attain liminf}\n\\lim\\limits_{j \\rightarrow \\infty} \\limsup_{T \\rightarrow \\infty} \\sup\\limits_{x \\in \\M_{\\epsilon}} \\mathbb{P}\\left[  \\Delta_{T,k,x,R_{j}} \\right] = 0.\n\\end{equation}\nWithout loss of generality, we can restrict ourselves to a regime of $R,T$ where there exists a large enough $C=C_{\\M} >0$ such that for all ratios $R\/T \\leq \\frac{inj(\\M)}{C}$, we have that $\\mbox{vol}_g B(x, R\/T) = \\pi \\frac{R^2}{T^2} + o(\\frac{R^2}{T^2})$.\n\nWe re-express\n\\begin{align} \\label{e:expect_int_geom}\n& {\\mathbf E} \\left[  \\int\\limits_{\\M} \\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+} \\, dx  \\right] \\\\\n\\nonumber & = \\int\\limits_{\\Omega} \\int\\limits_{\\M}  \\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+} \\, dx \\, d \\mathbb{P}(\\omega).\n\\end{align}\nAn application of Fubini's Theorem along with the measure bounds for the sets $\\Delta_{T,k,x,R}$ and $\\M_{\\epsilon}$ reduces bounding (\\ref{e:expect_int_geom}) to understanding the sum\n\\begin{equation*}\n\\left( \\int\\limits_{\\M} \\int\\limits_{\\Delta_{T,k,x,R}} + \\int\\limits_{\\M} \\int\\limits_{\\Omega \\cap \\Delta_{T,k,x,R}^{\\complement}} \\right) \\underbrace{\\left|\\frac{\\pazocal{N}_{V,norm}\\left( f_T,  k, x, (R+D)\/T  \\right)}{c_{2, \\alpha} \\cdot \\operatorname{Vol}(B(R+D))} - C_k  \\right|_{+}\\, d \\mathbb{P}(\\omega)   \\, dx }_{=: F \\, d \\mu} .\n\\end{equation*}\nThis the sum of integrals breaks up into a further $5$ integrals in total and we estimate each accordingly:\n\\begin{equation*}\n\\int\\limits_{\\M_{\\epsilon}} \\int\\limits_{\\Delta_{T,k,x,R}} F  \\, d \\mu \\leq \\xi^{-c_0} \\cdot \\sup\\limits_{x \\in \\M_{\\epsilon}} \\mathbb{P}[\\Delta_{T,k,x,R}],\n\\end{equation*}\n\\begin{equation*}\n\\int\\limits_{\\M\\setminus\\M_{\\epsilon}} \\int\\limits_{\\Delta_{T,k,x,R}} F  \\, d \\mu \\leq \\frac{\\epsilon}{5} \\cdot \\xi^{-c_0} \\cdot \\sup\\limits_{x \\in \\M_{\\epsilon}} \\mathbb{P}[\\Delta_{T,k,x,R}],\n\\end{equation*}\n\\begin{equation*}\n\\int\\limits_{\\M_{\\rho}} \\int\\limits_{\\Delta_{T,k,x,R}^{\\complement}} F  \\, d \\mu \\leq \\frac{\\epsilon}{5},\n\\end{equation*}\n\\begin{equation*}\n\\int\\limits_{  \\{V=0\\}^{\\sqrt{\\rho\/2}} } \\int\\limits_{\\Delta_{T,k,x,R}^{\\complement}} F  \\, d \\mu \\leq C_{\\M} \\cdot \\frac{\\epsilon^2}{50}, \\mbox{ thanks to the finiteness of } \\{V=0\\} \\mbox {, and }\n\\end{equation*}\n\\begin{equation*}\n\\int\\limits_{\\M\\setminus \\left( \\M_{\\rho} \\cup (\\{V=0\\}^{\\sqrt{\\rho\/2}} \\right)} \\int\\limits_{\\Delta_{T,k,x,R}^{\\complement}} F  \\, d \\mu \\leq C_{\\M} \\cdot \\frac{\\epsilon^2}{25}, \\mbox{ as }  \\left( \\M\\setminus  \\M_{\\rho} \\right) \\cap (\\{V=0\\}^{\\sqrt{\\rho\/2}}  \\neq \\emptyset \\mbox{ is possible}\n\\end{equation*}\nwhere $C_{\\M} > 0$ is an absolute constant depending only on the volume of $\\M$, which we have normalized to $1$.\n\nKeeping in mind that $\\epsilon$ was given to us, we restrict the auxiliary parameters introduced in our proof as follows: 1) $\\xi < (\\frac{\\epsilon}{5})^{1\/c_0}$, 2) take $R_0$ such that for all $R_{j}  \\geq R_0$ with $\\{R_{j}\\}_j$ as in \\eqref{eq:double limit R attain liminf} (which in turn, as in the proof of Theorem \\ref{t:conv_prob}, requires us to carefully select our parameters $\\beta_{1,1}, \\beta_{1,2}, \\beta_{1,3}, \\beta_2$ each of which themselves depends on $\\eta$), we know there exists $T_0(R_{j})$ (implicitly depending upon our string of stability parameters $\\beta$) such that for all $T \\geq T_0$ we have $ \\sup\\limits_{x \\in \\M_{\\rho}} \\mathbb{P}[\\Delta_{T,k,x,R_{j}}] < \\frac{\\epsilon \\xi^{c_0}}{5}$. We conclude that \\eqref{e:expect_int_geom} is bounded above by $\\epsilon$. This completes the proof of our proposition.\n\\end{proof}\n\nWe conclude by noting that our spectral parameter $T_0$ being dependent on the previous parameters $\\beta_{1,2}$ and $\\epsilon$ is a reflection of how our covering of $\\M \\setminus \\M_{\\epsilon}$, by geodesic disks of radius $R\/T$, is governed by the maximal order of vanishing amongst all zeroes of $V$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:Introduction}Introduction}\n\n\\section{Introduction} \n\nUltracold atomic gases confined in optical lattices with their unique controllability allow for artificial quantum engineering of lattice Hamiltonians with large system sizes beyond the computational reachability of classical simulations~\\cite{1998_Zoller_Jaksch_PRL,2002_Hofstetter_Cirac_PRL,2007_Lewenstein_AP,2008_Bloch_Dalibard_RMP,2010_Esslinger_CMP,2015_Lewenstein_RPP}. In the experiments, Bose and Fermi Hubbard models as well as spin Hamiltonians have  all been emulated to study both equilibrium quantum phase transitions of ground states and also  out-of-equilibrium many-body dynamics. Mott transitions of both bosons and fermions have been found in experiments~\\cite{2002_Greiner_Mandel_Nature,2008_Jordens_Strohmaier_Nature,2008_Schneider_Hackermuller_Science}, and even super-change mediated  magnetic phases~\\cite{2013_Greif_Uehlinger_Science,2015_Hart_Duarte_Nature,2016_Mazurenko_Chiu_arXiv} have recently been accomplished through the state-of-the-art quantum microscope techniques~\\cite{2015_Haller_Hudson_NatPhys,2015_Cheuk_Nichols_PRL,2015_Parsons_Huber_PRL,2015_Edge_Anderson_PRA,2015_Omran_Boll_PRL,2016_Greif_Parsons_Science,2016_Cheuk_Nichols_PRL,2016_Parsons_Mazurenko_Science,2016_Boll_Hilker_Science,2016_Cheuk_Nichols_Science,2016_Brown_Mitra_arXiv}.  Observing novel quantum dynamics in a strongly correlated setting is presently attracting growing experimental research interests in cold atoms and other synthetic quantum systems~\\cite{2012_Bloch_Light_Nature,2014_Zwierlein_PRL,2014_Monroe_Nature,2016_Greiner_Quantum_Science,2017_Nagerl_Bloch_Science,2017_Lukin_Observation_Nature,2017_Monroe_Observation_Nature,2017_Hulet_Formation_Science,2017_Everitt_observation}. \n\nThermalization---a most common phenomenon for interacting particles---could break down in isolated quantum systems subjected to random disorder potentials. \nStarting from Anderson's seminal work on localization~\\cite{1958_Anderson_Absence_PRL}, it has now been well-established that non-interacting particles moving in a disordered medium will be localized. \nThe robustness against interaction effects yet remained controversial until the recent studies of many-body localization~\\cite{Basko06,huse2015many,2015_Altman_review}. Through recent studies, the persistence of  localization in interacting systems has been established through a perturbative field theory calculation~\\cite{Basko06} and a rigorous mathematical proof~\\cite{imbrie2014many} together with extensive numerical works~\\cite{oganesyan2007,pal2010,moore2012,2013_Bauer_JSM,vadim,bardarson2014,2015_Singh_MBL_PRL,2015_Khemani_Pollmann_PRL,2015_Yu_Pekker_arXiv,2015_Lim_Sheng_PRB,2016_Kennes_Karrasch_PRB,2015_Bera_Many_PRL}. To describe the MBL phase, a phenomenological theory of local integrals of motion has been proposed, which provides a physical picture of highly constrained dynamics in the localized phase~\\cite{2014_Huse_MBL_PRB,2013_Serbyn_PRL,chandran2015,Ros2015420}. The consequent dynamical phenomena have been observed in experiments of ultracold atoms and trapped ions~\\cite{blochmbl,DemarcoDisorder,2015_Monroe_MBL_NatPhys,2016_Bordia_Bloch_MBL_PRL,2017_Bloch_Periodically_NatPhys,2016_Choi_Bloch_MBL_Science,2017_Bloch_Probing_arXiv,2017_Bloch_Signatures_PRX}. \n\nMore recent theoretical efforts in the MBL context are devoted to understanding the delocalization transition~\\cite{2015_Huse_Theory_PRX,2015_Potter_Universal_PRX,2017_Huse_Rare_AdP,2017_Huse_Critical_PRX} to the quantum thermal phase where eigenstate thermalization hypothesis~\\cite{1991_Deutsch_Quantum_PRA,1994_Srednicki_Chaos_PRE} is expected to hold, for that this type of transition does not have an analogue in the non-interacting problem of Anderson localization. To characterize the transition, various diagnostics such as entanglement entropy scaling and many-body energy level statistics have thoroughly  been investigated in theory, and scaling functions based on these quantities have been proposed to describe the MBL-to-ergodic criticality. Across the transition the quantum entanglement entropy scaling would switch from area- to volume-law~\\cite{2013_Bauer_JSM}. The level statistics exhibits a transition from Poisson type to Wigner-Dyson~\\cite{oganesyan2007}. These quantum entanglement and statistical quantities offer concrete measures to describe the criticality,  but these theoretical ``observables\"  turn out to be extremely challenging to measure, and it remains unclear how to accurately probe MBL localization length in experiments. The present theoretical debate about the validity of Harris-Chayes bound~\\cite{1974_Harris_Effect_JPC,1986_Chayes_Finite_PRL} for MBL transition~\\cite{2014_Kjall_MBL_PRL,2015_Huse_Theory_PRX,2015_Potter_Universal_PRX,2015_Luitz_MBL_PRB}  makes the task to probe localization length exceedingly desirable. \n\n\nHere we propose to use quench dynamics to probe the localization length of interacting atoms in disorder potential and the corresponding MBL criticality. \nWe find that an added hole in the MBL phase shows an exponentially decaying density profile after long-time dynamical evolution, whereas in the ergodic phase its density distribution completely spreads over the whole lattice system. The size of the density profile in the long-time-evolved final state defines a localization length, whose critical behavior can be directly probed in experiments. \nThe transition point determined by our defined localization length agrees with  that by other diagnostics. \nIn our numerical results, the critical exponent is found to violate Harris-Chayes bound~\\cite{1974_Harris_Effect_JPC,1986_Chayes_Finite_PRL}, \nwhich implies many-body localization criticality is beyond the description of conventional field theory or renormalization group study for disorder systems.\n It is worth remarking here that this work is rather to propose an experimental scheme to detect the MBL localization length and criticality than to calculate a precise critical exponent. Whether the Harris-Chayes bound holds or not at the MBL transition would rely on future experiments. \nThe proposed schemes are justified by benchmarking to  the extensively studied 3D Anderson localization.\n\n\n\n\n\n\n\n\n\n\\section{Model and Method}\n\nTo be concrete we consider a model Hamiltonian of spinless fermions with nearest neighbor interactions, \n\\begin{equation} \\label{Hamiltonian}\n\\hat{H} = \\sum_{<jj'>}\\left[-t\\left(\\hat{c}_{j}^{\\dagger}\\hat{c}_{j'}+h.c.\\right)+V\\hat{n}_{j}\\hat{n}_{j'}\\right] + \\sum_{j}h_{j} \\hat{n}_{j}.  \n\\end{equation}\nHere $<jj'>$ denotes nearest-neighboring sites, $t$ is the tunneling matrix element ($t$ is set to be the energy unit throughout), $\\hat{c}_{j}^{\\dagger}$ ($\\hat{c}_{j}$)  is a  fermionic creation (annihilation) operator. The second term describes the interaction between nearest-neighboring sites with $V$  the interaction strength and $\\hat{n}_{j} = \\hat{c}_{j}^{\\dagger}\\hat{c}_{j}$ is the number operator. The last term corresponds to the disorder potential, where $h_{i}$ is drawn from uniform distribution $\\left[-W,W\\right]$. This model is equivalent to the spin-1\/2 XXZ model with random field via Jordan-Wigner transformation.\nWe consider this model instead of the experimental system of spinful fermions for the interests of performing numerical calculations of large system sizes. The proposed method and the findings based on the spinless fermion model to present below are also expected to hold for spinful fermions as well due to the robust universality of the MBL transition. We focus on half filling in this study. \n\n\n We propose to use quench dynamics to probe MBL criticality, which is experimentally accessible with quantum microscope techniques in ultra-cold atomic gases. As elaborated in previous studies, interacting fermions will display many-body localization with strong disorder potential. In analogy to Anderson localization, the response to a local quench is expected to be bounded within a local region restricted by the localization length~\\cite{2017_Deng_Log_PRB}.  We thus propose to measure the localization length by monitoring the perturbed density profile  after a local quantum quench. \n\nMore precisely, the initial state we choose is a ``charge density wave (CDW) state\" with atoms occupying every other lattice site, which is the same as used in the experiment~\\cite{blochmbl}. We average over two different types of CDW states, atoms occupying either all even sites or odd sites. \nThen we let the state evolve for long enough time say $\\tau_1$ such that the degrees of freedom would ``locally equilibrate\" with each other. \n \nThen we introduce a sudden local quench. We provide two quench protocols for comparison---(I) lowering the potential of the quench site to zero, that had much higher energy than other sites and was initially empty before the quantum quench; and (II) performing a measurement on the quench site and remove the atom at this site. The details of the quench protocols are to be specified in Section~\\ref{sec:quenchresults}. \nThe dynamics following quench protocol-(I) is completely unitary and is thus more convenient for theoretical analysis, whereas the dynamics in protocol-(II) is non-unitary because of the involved measurement, yet has the advantage of being more experimentally feasible with quantum microscope techniques~\\cite{2015_Haller_Hudson_NatPhys,2015_Cheuk_Nichols_PRL,2015_Parsons_Huber_PRL,2015_Edge_Anderson_PRA,2015_Omran_Boll_PRL,2016_Greif_Parsons_Science,2016_Cheuk_Nichols_PRL,2016_Parsons_Mazurenko_Science,2016_Boll_Hilker_Science,2016_Cheuk_Nichols_Science,2016_Brown_Mitra_arXiv}. We will provide theoretical analysis based on the unitary evolution following quench protocol-(I) in this section, and provide simulated numerical results for both protocols in Section~\\ref{sec:quenchresults}. \n\n\n\n\nDespite the difference, both quench protocols lead to a hole on the quench site in the density profile. \nThen the density profile of this added hole is monitored. This density profile is expected to be localized (delocalized) in the MBL (ergodic) phase. The localization length can be correspondingly extracted from the time-evolved density profile at long-time limit say at $\\tau_2$. It is worth emphasizing here that our proposed scheme to probe MBL criticality is rather easily adaptable to specific experimental setups instead of being restricted to the particular local quench protocol considered here. \n\n\n\n\n\n To make it quantitative, here we analyze the quench dynamics following protocol-(I). We have total number of $L+1$ sites labeled from $0$ to $L$. The potential energy at site-$0$ is set to be  much higher  than other sites before quench, so is initially empty. After the quench this site gets filled in dynamics because its potential is then lowered down. Denoting the pre- and post- quench Hamiltonian as $H_0$ and $H$,  the perturbed density profile measures\n$\n\\delta n_j = \\langle \\psi(\\tau_1) |  {\\cal O}_j  |\\psi(\\tau_1) \\rangle \n$   \nwhere we have \n$$\n{ {\\cal O}_j} = e^{iH_0 \\Delta \\tau} \\hat{n}_j e^{-iH_0 \\Delta \\tau } - e^{iH \\Delta \\tau} \\hat{n}_j e^{-iH \\Delta \\tau }, \n$$\nwith $\\Delta \\tau = \\tau_2 - \\tau_1$. \nThe difference between $H$ and $H_0$ is strictly local near site-$0$. In the MBL phase, the phenomenological theory of local integrals of motion~\\cite{2014_Huse_MBL_PRB,2013_Serbyn_PRL,chandran2015,Ros2015420} implies that \nthe operator norm of ${\\cal O}_j$ decays exponentially  \n\\begin{equation} \n\\overline{ || {\\cal O} _j ||}  \\propto e^{-d_j \/\\xi}  \n\\end{equation} \nwith $\\xi$ a localization length, and $d_j $ the distance of the $j$-th lattice site to site-$0$, which is given by  $d_j = {\\rm min}(j, L+1-j)$ in a periodic boundary system with  size $L+1$. In the thermal phase, the support of ${\\cal O} _j$ would spread over the whole system through a linear light-cone dynamics. ~\\cite{2012_Bloch_Light_Nature, 2017_Deng_Log_PRB, Lieb1972}\nWe can thus extract the localization length from the perturbed density profile $\\delta n_j$ according to  \n\\begin{equation} \n\\xi = \\frac{\\sum_j  d_j \\overline{ \\delta  n_j} }{\\sum_j \\overline{ \\delta n_j}}. \n\\end{equation}\nIn our numerical results shown in Fig.\\ref{fig:Figure1}, we find stronger disorder strength makes the density profile more  localized and thus  $\\xi$ smaller.  Increasing interaction strength makes the density profile more extended, and  $\\xi$ becomes larger.\n\n\n\n\n\n\n\n\n\nThe system is completely localized at the strong disorder limit, whose eigenstates are simply product states. In the deep localized phase, the localization length is vanishing in our definition. Upon decreasing disorder, the localization length becomes larger but still remains to be a finite number in the MBL phase, i.e., with the scaling form $L^0$ as we change the system size. Further decreasing disorder, the scaling form of the localization length switch to $L\/4$ immediately after the ergodic transition happens. In the one dimensional system, the scaling behavior of $\\xi$ would resemble the entanglement entropy in both the localized and the ergodic phases. The difference is that $\\xi$ can be measured in experiments whereas to measure the entanglement entropy is difficult. \n \nAssuming a unique diverging length scale $\\delta W^{-\\nu}$ at  the MBL transition leads to a natural finite-size scaling ansatz for the localization length\n\\begin{equation} \\label{eq:scaling}\n\\xi\/L \\sim  g(L^{1\/\\nu} \\delta W)\n\\end{equation} \nwith $\\delta W = W - W_c$. \n \n This scaling ansatz is consistent with the fact that $\\xi\/L$ has a jump across the MBL-to-ergodic transition in the thermodynamic limit. \n\nGiven the scaling form of $\\xi\/L$, the transition point can be extracted from the crossing of the curves for $\\xi\/L $ versus the disorder strength $W$ with different system sizes. \nThe critical exponent $\\nu$ as well as critical value $W_c$ are determined by finite-size scaling analysis~\\cite{2014_Kjall_MBL_PRL,2015_Luitz_MBL_PRB}.  \n\n\n\\begin{figure}[htp]\n\t\\includegraphics[width=0.8\\linewidth]{Figure1_a}\n\t\\includegraphics[width=0.8\\linewidth]{Figure1_b}\n\t\\caption{(Color online)\n          The perturbed density profile after a local quench following protocol-(I) (see main text). \n\tHere we choose the system size $L = 16$, and average over $1000$ disorder realizations in this figure. \n\t(a), The density profile distribution with varying interaction strengths with disorder strength fixed at $W\/t = 9$. \n\t(b), The distribution for different disorder strengths with interaction fixed at $V\/t =1 $.}\n\\label{fig:Figure1} \n\\end{figure}\t\n\n\n\\begin{figure}[htp]\n\t\\includegraphics[angle=0,width=0.8\\linewidth]{Figure2_a}\n\t\\quad\n\t\\includegraphics[angle=0,width=0.8\\linewidth]{Figure2_b}\n\t\\quad\n\t\\includegraphics[angle=0,width=0.8\\linewidth]{Figure2_c}\n\t\\caption{\\label{fig:Figure2} (Color online) \nMany-body localization criticality from our proposed quench dynamics with protocol-(I) (see main text). (a) and (b) MBL length as a function of disorder strength for interaction strength $V\/t = 1$ and $V\/t = 1.5$ respectively. The curve intersects at $W_{c}\/t = 4.90 \\pm 0.03$ for $V\/t = 1$ and at $W_{c}\/t = 5.57 \\pm 0.04$ for $V\/t =1.5$. (c) Finite size scaling with scaling function $g\\left( L^{1\/\\nu}\\delta W\\right )$ where $\\delta W = W - W_{c}$. Critical exponent is estimated to be $\\nu = 1.02 \\pm 0.09$.}\n\\end{figure}\n\n\n\n\\section{Simulated results for post-quench dynamics} \n\\label{sec:quenchresults} \n\n \nThe proposed dynamical experiment is simulated by performing exact diagonalization calculation. We use periodic boundary condition to minimize finite-size effects in the numerical calculation, and the system sizes are $L = 8, 10, \\ldots 16$. For the experimental case, the finite-size effects are not expected to be too significant for that the number of sites in optical lattice experiments could reach the order of one hundred, much larger than that can be numerically simulated. \n\n\n\\subsection{Quench Protocol-(I) } \n\nIn the quench protocol-(I), we have an empty site labeled by an index $0$, that has much higher potential energy than other lattice sites in the system. \nThe pre-quench lattice sites are labeled as $1, 2, \\ldots L$. In the numerical simulation of pre-quench dynamics, we restrict to the Hilbert space setting the particle number at site $0$ to be explicitly $0$. \nAt $\\tau_1$ the potential at site-$0$ is lowered down to the same level as other lattice sites, which instantaneously creates a hole in the system at site $0$. The system is then evolved with a long time $\\tau_2$, where the created hole eventually stabilizes by interacting with other particles. \nFor $L \\le 12$, we perform the full matrix diagonalization and average over $10^4$ disorder realizations. For $L \\ge 14$, a  Krylov space expansion is implemented for the unitary $e^{-iHt}$ in order to save cost on memory, and we average over $10^3$ disorder realizations. \nThe dynamical evolution of local density $\\delta n_{j} = n_{j}\\left(\\tau_1\\right) - n_{j}\\left(\\tau_2\\right)$  is calculated, $n_j (\\tau) = \\langle \\psi (\\tau) | \\hat{n}_j |\\psi(\\tau)\\rangle$, with $|\\psi(\\tau)\\rangle$ the time-dependent quantum state. \n\n\nHere we would like to describe a special treatment on the perturbed density profile at the quenched site, i.e. $\\delta n_0$.  If we are in the thermal phase, the final state in our proposed dynamical procedure is ergodic, and the density distribution is uniform on every lattice site. \n\nSince at time $\\tau_1$ the occupation number at site-$0$ is different from other sites, \n$\\delta  n_0$ is compensated by adding $1\/2$.  This compensation makes $\\delta n_j$ a flat profile in the thermal phase. Meantime, the so-defined $\\delta n_j$ then obeys a normalization condition  $\\sum_j \\delta n_j = 1\/2$. \n \n\n\n\nFrom Fig.~\\ref{fig:Figure1}, averaged over different disorder realizations, $\\overline {\\delta n_i}$ indeed shows exponential decay in the MBL phase with strong disorder, and becomes flat in the ergodic phase at weak disorder. Note that the notation $\\overline{\\ldots}$ implies averaging over disorder throughout. The interaction dependence is also studied. Increasing interaction strength in the localized phase only makes the density profile of the hole a bit more delocalized, and it still shows an exponential decay, showing that the signature of many-body localization in quench dynamics is stable against interaction effects. \n\n\nFig.~\\ref{fig:Figure2} (a) shows the many-body localization criticality exhibited by the hole-profile localization length in our proposed quench dynamics. \nWith an interaction strength $V\/t = 1$,  the extracted critical disorder strength from the localization length is found to locate at $W_{c}\/t = 4.90 \\pm 0.03$.  As we increase the interaction strength, the localization becomes less robust and the required disorder strength to stabilize MBL gets larger (compare Figs.~\\ref{fig:Figure2} (a, b)). To find out the critical exponent $\\nu$, we collapse the data to the  scaling function $g\\left(L^{1\/\\nu}\\delta W\\right)$, which gives an estimate that $\\nu = 1.02 \\pm 0.09$. This value breaks the Harris-Chayes criterion $\\nu \\ge 2\/d$ with $d$ the spatial dimension~\\cite{1974_Harris_Effect_JPC,1986_Chayes_Finite_PRL}.  \n This violation would imply many-body localization criticality is beyond the description of conventional field theory or renormalization group study for disorder systems. \n\nWith the numerical results, we explicitly confirm that the proposed quench dynamics can be used to study localization length and MBL criticality. \n\n\n\n\n\\subsection{Quench Protocol---(II)} \n\n\n \nIn the quench protocol-(II), we still let the initial state evolve for long enough time $\\tau_1$. The quench is performed in a different way from protocol-(I). At $\\tau_1$, we  perform a  conditional measurement on a given site, here labeled as site-$1$. Then we do a post-selection, where measurement-outcome state is discarded if there is no particle on the measured lattice site, and the state is kept otherwise. For the kept state, we remove the particle at this site to create a hole in the density profile. This process can be carried out by quantum microscope techniques in a straightforward way. For the initial state, we choose both CDW and random states---the random state case is used as a comparison to determine whether there is any artifact due to special choice of CDW states. CDW initial states have been used in cold atom experiments to study many-body localization~\\cite{blochmbl}. \n\n\\begin{figure}[htp]\n\t\\centering\n\t\\includegraphics[angle=0,width=0.8\\linewidth]{Figure3}\n\t\\caption{(Color online)\n\t\tThe density profile after removing a particle on site-$1$ for different disorder strengths following quench protocol-(II) (see main text). We choose  a system size $L = 16$, and an interaction strength $V\/t = 1$, and average over 1000 disorder realizations in this figure. The density profile in this plot is normalized to $1$. }\n\t\\label{fig:Figure3} \n\\end{figure}\n\nWe monitor the post-quench density profile of the hole $\\delta n_j$ at a long time $\\tau_{2}$. \nFrom Fig.~\\ref{fig:Figure3}, we confirm that the hole-profile is localized (extended) in MBL (ergodic) phase. \nThe local hole density is defined as $\\delta n_{j} = 1\/2 - n_{j}\\left(\\tau_{2}\\right)$, and it obeys a normalization condition $\\sum_j \\delta n_j = 1$. \nHere, we remark that removing one particle is necessary in this quench protocol to study MBL localization length---a conditional measurement  without removing the particle could not create a well-defined hole in the density profile. \n\nThe localization length extracted from this protocol is shown in Fig.~\\ref{fig:Figure4}.  We find its behavior is similar to the results for quench protocol-(I) (see Fig.~\\ref{fig:Figure2}). \nIn comparison of  CDW with random initial states,  we find no qualitative difference.  For  these two different choices of inital states, the crossing point of MBL localization length  versus disorder strength with different system sizes is consistent with each other. We find systematic data-collapse using the scaling form in Eq.~\\eqref{eq:scaling} for both of them.    \nThis implies the MBL criticality can be studied by using CDW initial states which is relatively simpler to implement in optical lattice experiments. \n\n\n\nComparing the results in  Fig.~\\ref{fig:Figure2} and Fig.~\\ref{fig:Figure4}, it is evident that quench protocol-(II) works as well as quench protocol-(I), although the dynamics is non-unitary for (II) but unitary for (I). The similarity suggests that the details of the quench protocol are not crucial for  the study of localization length and MBL criticality, provided that the quench creates a well-defined hole in the density profile. \n\n\n\n \n \nIn our proposal, it is crucial to know the required dynamical time scales before and after the quench, i.e., $\\tau_1$ and $\\tau_2$.  \nFor the dynamics to reveal intrinsic MBL physics, it is necessary that the system stabilizes for both before and after the quantum quench. \nIn Fig.~\\ref{fig:Figure5},  we show details of the dynamical evolution. \nFig.~\\ref{fig:Figure5} (a, c) show the pre-quench dynamics with $V\/t = 1$ and $1.5$. We take one type of CDW state with atoms occupying all odd sites and monitor the atom number imbalance  as $I = \\frac{N_{o}-N_{e}}{N_{o}+N_{e}}$, with $N_o$ and $N_e$ the total particle numbers on odd and even lattice sites.\n It can be seen that the system stabilizes after about $20$ times of tunneling time. \nFig.~\\ref{fig:Figure5} (b, d) show the post-quench dynamics in the hole density profile $\\delta n_j$. \nThe system is found to stabilize after about $10$ times tunneling time. \nTaking the two steps into account, the required time scale to perform the quench experiment is around $30$ times tunneling time, which is about $30-100$ milliseconds for a typical optical lattice with  tunneling time around one millisecond~\\cite{2008_Bloch_Dalibard_RMP}. This is reasonably within the lifetime of cold atom experiments. \n \n \\begin{figure}[H]\n \t\\centering\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_a}\n \t\\end{minipage}}\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_b}\n \t\\end{minipage}}\n \t\\quad\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_c}\n \t\\end{minipage}}\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_d}\n \t\\end{minipage}}\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_e}\n \t\\end{minipage}}\n \t\\subfigure[]{\n \t\t\\begin{minipage}{0.48\\linewidth}\n \t\t\t\\centering\n \t\t\t\\includegraphics[width=1.7in]{Figure4_f}\n \t\\end{minipage}}\n \t\n \t\\caption{ (Color online) \n \t\tMany-body localization criticality out of the quench protocol-(II) (see main text). (a, b, c, d) show the localization length versus disorder strength with the interaction fixed at $V\/t = 1$. The transition point and criticality is estimated to be $W_{c}\/t = 4.08 \\pm 0.04$ and $\\nu = 0.86 \\pm 0.06$ for CDW initial states and $W_{c}\/t = 3.82 \\pm 0.04$ and $\\nu = 0.73 \\pm 0.06$ for random initial state. In (e) and (f), interaction strength is $V\/t = 1.5$. The transition point and criticality is estimated to be $W_{c}\/t = 4.69 \\pm 0.05$ and $\\nu = 0.92 \\pm 0.08$. In the calculation, for $L \\le 12$, the full matrix diagonalization is performed and the results are averaged over $10^4$ disorder realizations. For $L \\ge 14$, a Krylov space expansion is implemented for the unitary $e^{-iHt}$, and we average over $10^3$ disorder realizations.  }\n \t\\label{fig:Figure4}\n \\end{figure}\n\n\\begin{figure}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure5_a}\n\t\\end{minipage}}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure5_b}\n\t\\end{minipage}}\n\t\\quad\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure5_c}\n\t\\end{minipage}}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure5_d}\n\t\\end{minipage}}\n\t\\caption{\\label{fig:Figure5} \nPre- and post-quench dynamics following the quench protocol-(II). Here we take CDW initial states, and choose $L = 12$. (a) and (c) correspond to pre-quench dynamics in number imbalance $I$ (see main text) with $V\/t = 1$ and $1.5$ respectively. \n(b) and (d) show the post-quench dynamics in the density profile $\\delta n_2$ for interaction strengths $V\/t = 1$ and $1.5$.}\n\\end{figure}\n\n\n\\subsection{Interaction dependence of the critical disorder strength from local quench dynamics} \nFig.~\\ref{fig:Figure7} shows a systematic study of interaction effects on MBL transition for both quench protocols.  At the interaction strength $V\/t = 2$, the fermion lattice model maps onto  random field Heisenberg model which has been extensively studied in the literature. The transition point determined from our proposed dynamical experiment agrees with previous studies using other diagnostics~\\cite{oganesyan2007,pal2010,moore2012,2013_Bauer_JSM,vadim,bardarson2014,2015_Singh_MBL_PRL,2015_Khemani_Pollmann_PRL,2015_Yu_Pekker_arXiv,2015_Lim_Sheng_PRB,2016_Kennes_Karrasch_PRB}. Our approach has an advantage in that all the quantities required to extract the localization transition and criticality can be probed directly in optical lattice experiments. \n\n\nIt is worth noting that for large system size (see Fig.\\ref{fig:Figure2}), the data crossing to determine the localization transition point has a slight rightward drift as a result of finite-size effect. To reach a conclusive answer for MBL criticality would rely on the experiments which can go to large system sizes beyond the computation capability of numerical simulations with classical resources. The advantage of our proposed strategy to probe MBL criticality is that the required ingredients are all presently accessible with cold atoms in optical lattices. \n\n\\begin{figure} [htp]\n\t\\includegraphics[angle=0,width=.7\\linewidth]{Figure6}\n\t\\caption{\\label{fig:Figure7} {{Critical disorder strength as a function of interaction strength as extracted from our proposed local quench dynamics.}}}\n\\end{figure}\n\n\n\\section{Benchmarking the dynamical protocol with Anderson localization}\n\nTo benchmark the method of probing the many-body localization length in our study,  we carry out a simulation for the well-understood three-dimensional Anderson localization whose Hamiltonian reads, \n\\begin{equation} \\label{Hamiltonian2}\n\\hat{H} = \\sum_{<jj'>}-t\\left(\\hat{c}_{j}^{\\dagger}\\hat{c}_{j'}+h.c.\\right) + \\sum_{j}h_{j}\\hat{n}_{j}. \n\\end{equation} \nFor this non-interacting model, we extract the localization length from the long-time-evolved density profile of a single particle initialized at one lattice site. This single-particle  density profile indeed takes an exponential decay form. For $L = 16$ ($20$, $24$ and $32$), the density profile is averaged over $1000$ ($100$) realizations and in the vicinity of expected transition interval, we average over 1000 realizations for all sizes. \nBy calculating the localization length from the density profile, we find $W_c\/t = 15.88 \\pm 0.14$ (see Fig.~\\ref{fig:Figure8}) and the critical exponent $\\nu = 1.6 \\pm 0.2$, which are consistent with well-known results for the Anderson model~\\cite{1999_Slevin_Corrections_PRL}.  This justifies the validity of our proposed dynamical protocols to extract localization criticality. \n\n\n\n\\begin{figure} [H]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure7_a}\n\t\\end{minipage}}\n\t\\subfigure[]{\n\t\t\\begin{minipage}{0.48\\linewidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1.7in]{Figure7_b}\n\t\\end{minipage}}\n\t\\caption{\\label{fig:Figure8} (Color online) (a) Localization length as a function of disorder strength for the three dimensional Anderson model to benchmark our method. The curves for different system sizes intersects at $W_c\/t = 15.88 \\pm 0.14$ . (b) Data collapse to a  scaling function $g_{\\rm AL} \\left[L^{1\/\\nu}\\delta\\right]$ where $\\delta = W - W_{c}$, which leads to  $\\nu = 1.6 \\pm 0.2$.}\n\\end{figure}\t\n\n\n\n\\section{Conclusion} \n\nIn summary, we propose to use local quench dynamics to probe the MBL-to-ergodic criticality. Its validity is confirmed by benchmarking towards  the well-known 3D Anderson localization. In our proposed schemes, the localization length could be extracted from the exponential decay of a perturbed density profile after a local quantum quench. This proposal is expected to guide future experiments in probing MBL criticality with ultracold atoms in disordered optical lattices. Moreover, we provide a finite-size scaling form of the localization length across the transition, which is directly applicable in analyzing the quench-dynamics data out of the proposed experiments. It is worth future study to sort out the finite-size effects in the quench dynamics of many-body localization, in particular about violation of Harris-Chayes bound, for example by developing dynamical renormalization group techniques.\n\n\\section{Acknowledgments}\n{We thank I. Bloch, Y. Takahashi, D.A. Huse, E. Altman, G. Refael, X. Chen, J. Alicea, L. Jiang and S. M. Girvin for helpful discussions. \nThis work is supported by National Program on Key Basic Research Project of China under Grant No. 2017YFA0304204, National Natural Science Foundation of China under Grants No. 117740067,  and the Thousand-Youth-Talent Program of China. \n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}