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Alexander and Orbach (AO) in 1982 conjectured that the simple random walk on critical percolation clusters (also known as the ant in the labyrinth) in Euclidean lattices exhibit mean field behavior; for instance, its spectral dimension is $4/3$. While false in low dimensions, this is expected to be true above the upper critical dimension of six. First rigorous results in this direction go back to Kesten who verified this on the tree. After many developments, in a breakthrough work, Kozma and Nachmias [KN] established the AO conjecture for bond percolation on $\mathbb{Z}^d$ for $d>19$ and $d>6$ for the spread out lattice. We investigate the validity of the AO conjecture for the critical level set of the Gaussian Free Field (GFF), a canonical dependent percolation model of central importance. In an influential work, Lupu proved that for the cable graph of $\mathbb{Z}^d$ (which is obtained by also including the edges), the signed clusters of the associated GFF are given by the corresponding clusters in a Poisson loop soup, thus reducing the analysis to the study of the latter. In 2021, Werner put forth an evocative picture for the critical behavior in this setting drawing an analogy with the usual bond case. Building on this, Cai and Ding established the universality of the extrinsic one arm exponent for all $d > 6.$ In this article, we carry this program further, and consider the random walk on sub-sequential limits of the cluster of the origin conditioned to contain far away points. These form candidates for the Incipient Infinite Cluster (IIC), first introduced by Kesten in the planar case. Inspired by the program of [KN], introducing several novel ideas to tackle the long range nature of this model and its effect on the intrinsic geometry of the percolation cluster, we establish that the AO conjecture indeed holds for any sub-sequential IIC, for all large enough dimensions., Comment: 76 pages, 15 figures

We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimization approach introduced for the local case, we construct global-time weak solutions and give a probabilistic interpretation for the solutions. An important ingredient in our analysis is a probabilistic interpretation of the enthalpy and temperature variables in terms of a particle system. Our approach in particular establishes the connection between the parabolic obstacle problem and the Stefan Problem for the nonlocal diffusions. For the melting problem, we show that our solution coincides with those studied in the literature., Comment: 42 Pages. Fixed some typos

Isoperimetric profile describes the minimal boundary size of a set with a prescribed volume. Itai Benjamini conjectured that the isoperimetric profile of the giant component in supercritical percolation experiences an averaging effect and satisfies the law of large numbers. This conjecture was settled by Biskup-Louidor-Procaccia-Rosenthal for 2D percolation, and later resolved by Gold for higher-dimensional lattices. However, more refined properties of the isoperimetric profile, such as fluctuations and large deviations, remain unknown. In this paper, we determine the large deviation probabilities of the isoperimetric constant in 2D supercritical percolation, answering the question by Biskup-Louidor-Procaccia-Rosenthal. Interestingly, while the large deviation probability is of surface order in the entire upper tail regime, a phase transition occurs in the lower tail regime, exhibiting both surface and volume order large deviations., Comment: 35 pages, 2 figures

Analyzing the spectral behavior of random matrices with dependency among entries is a challenging problem. The adjacency matrix of the random $d$-regular graph is a prominent example that has attracted immense interest. A crucial spectral observable is the extremal eigenvalue, which reveals useful geometric properties of the graph. According to the Alon's conjecture, which was verified by Friedman, the (nontrivial) extremal eigenvalue of the random $d$-regular graph is approximately $2\sqrt{d-1}$. In the present paper, we analyze the extremal spectrum of the random $d$-regular graph (with $d\ge 3$ fixed) equipped with random edge-weights, and precisely describe its phase transition behavior with respect to the tail of edge-weights. In addition, we establish that the extremal eigenvector is always localized, showing a sharp contrast to the unweighted case where all eigenvectors are delocalized. Our method is robust and inspired by a sparsification technique developed in the context of Erd\H{o}s-R\'{e}nyi graphs (Ganguly and Nam, '22), which can also be applied to analyze the spectrum of general random matrices whose entries are dependent., Comment: 36 pages

Many low temperature disordered systems are expected to exhibit Poisson-Dirichlet (PD) statistics. In this paper, we focus on the case when the underlying disorder is a logarithmically correlated Gaussian process $\phi_N$ on the box $[-N,N]^d\subset\mathbb{Z}^d$. Canonical examples include branching random walk, $*$-scale invariant fields, with the central example being the two dimensional Gaussian free field (GFF), a universal scaling limit of a wide range of statistical mechanics models. The corresponding Gibbs measure obtained by exponentiating $\beta$ (inverse temperature) times $\phi_N$ is a discrete version of the Gaussian multiplicative chaos (GMC) famously constructed by Kahane. In the low temperature or supercritical regime, the GMC is expected to exhibit atomic behavior on suitable renormalization, dictated by the extremal statistics of $\phi_N$. Moreover, it is predicted, going back to a conjecture made in 2001 by Carpentier and Le Doussal, that the weights of this atomic GMC has a PD distribution. In a series of works, Biskup and Louidor carried out a comprehensive study of the near maxima of the 2D GFF, and established the conjectured PD behavior throughout the super-critical regime. In another direction, Ding, Roy and Zeitouni established universal behavior of the maximum for a general class of log-correlated Gaussian fields. In this paper we continue this program simply under the assumption of log-correlation and nothing further. We prove that the GMC concentrates on an $O(1)$ neighborhood of the local extrema and the PD prediction holds, in any dimension $d$, throughout the supercritical regime, significantly generalizing past results. Unlike for the 2D GFF, in absence of any Markovian structure for general Gaussian fields, we develop and use as our key input a sharp estimate of the size of level sets, which could have other applications., Comment: 78 pages; new title, explanations and references added

Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices is formed by the adjacency matrix of an Erd\H{o}s-R\'{e}nyi graph $\mathcal{G}_{n,p}$ equipped with i.i.d. edge-weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. We focus on the case $p = \frac{d}{n}$, where most known techniques break down. So far, results were known only for $\mathcal{G}_{n,\frac{d}{n}}$ without edge-weights (Krivelevich and Sudakov, '03), (Bhattacharya, Bhattacharya, and Ganguly, '21) and with Gaussian edge-weights (Ganguly and Nam, '21). In the present article, we consider the effect of general weight distributions. More specifically, we consider the entries whose tail probabilities decay at rate $e^{-t^\alpha}$ with $\alpha>0$, where the regimes $0<\alpha<2$ and $\alpha>2$ correspond to tails heavier and lighter than the Gaussian tail respectively. While in many natural settings the large deviations behavior is expected to depend crucially on the entry distribution, we establish a surprising and rare universal behavior showing that this is not the case when $\alpha > 2.$ In contrast, in the $\alpha< 2$ case, the large deviation rate function is no longer universal and is given by the solution to a variational problem, the description of which involves a generalization of the Motzkin-Straus theorem, a classical result from spectral graph theory. As a byproduct of our large deviation results, we also establish new law of large numbers results for the largest eigenvalue. In particular, we show that the typical value of the largest eigenvalue exhibits a phase transition at $\alpha = 2$, i.e. the Gaussian distribution., Comment: 52 pages

Let $N$ be the number of triangles in an Erd\H{o}s-R\'enyi graph $\mathcal{G}(n,p)$ on $n$ vertices with edge density $p=d/n,$ where $d>0$ is a fixed constant. It is well known that $N$ weakly converges to the Poisson distribution with mean ${d^3}/{6}$ as $n\rightarrow \infty$. We address the upper tail problem for $N,$ namely, we investigate how fast $k$ must grow, so that the probability of $\{N\ge k\}$ is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by Poisson behavior only when $k^{1/3} \log k< (\frac{3}{\sqrt{2}})^{2/3} \log n$ (sub-critical regime) as well as pin down the tail behavior when $k^{1/3} \log k> (\frac{3}{\sqrt{2}})^{2/3} \log n$ (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost $k$ vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately $(6k)^{1/3}$. This settles the long-standing upper-tail problem in this case, answering a question of Aldous, complementing a long sequence of works, spanning multiple decades, culminating in (Harel, Moussat, Samotij,'19) which analyzed the problem only in the regime $p\gg \frac{1}{n}.$ The proofs rely on several novel graph theoretical results which could have other applications., Comment: 32 pages, 2 figures

We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on $\mathbb{R}^m$ are replaced by non-commutative laws of $m$-tuples. We prove an analog of the Monge-Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu's non-commutative $L^2$-Wasserstein distance using a new type of convex functions. As a consequence, we show that if $(X,Y)$ is a pair of optimally coupled $m$-tuples of non-commutative random variables in a tracial $\mathrm{W}^*$-algebra $\mathcal{A}$, then $\mathrm{W}^*((1 - t)X + tY) = \mathrm{W}^*(X,Y)$ for all $t \in (0,1)$. Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of $m$-tuples is not separable with respect to the Wasserstein distance for $m > 1$., Comment: 60 pages, revised

Large deviation behavior of the largest eigenvalue $\lambda_1$ of Gaussian networks (Erd\H{o}s-R\'enyi random graphs $\mathcal{G}_{n,p}$ with i.i.d. Gaussian weights on the edges) has been the topic of considerable interest. Recently in [6,30], a powerful approach was introduced based on tilting measures by suitable spherical integrals, particularly establishing a non-universal large deviation behavior for fixed $p<1$ compared to the standard Gaussian ($p=1$) case. The case when $p\to 0$ was however completely left open with one expecting the dense behavior to hold only until the average degree is logarithmic in $n$. In this article we focus on the case of constant average degree i.e., $p=\frac{d}{n}$. We prove the following results towards a precise understanding of the large deviation behavior in this setting. 1. (Upper tail probabilities): For $\delta>0,$ we pin down the exact exponent $\psi(\delta)$ such that $$\mathbb{P}(\lambda_1\ge \sqrt{2(1+\delta)\log n})=n^{-\psi(\delta)+o(1)}.$$ Further, we show that conditioned on the upper tail event, with high probability, a unique maximal clique emerges with a very precise $\delta$ dependent size (takes either one or two possible values) and the Gaussian weights are uniformly high in absolute value on the edges in the clique. Finally, we also prove an optimal localization result for the leading eigenvector, showing that it allocates most of its mass on the aforementioned clique which is spread uniformly across its vertices. 2. (Lower tail probabilities): The exact stretched exponential behavior of $\mathbb{P}(\lambda_1\le \sqrt{2(1-\delta)\log n})$ is also established. As an immediate corollary, we get $\lambda_1 \approx \sqrt{2 \log n}$ typically, a result that surprisingly appears to be new. A key ingredient is an extremal spectral theory for weighted graphs obtained via the classical Motzkin-Straus theorem., Comment: 49 pages

We derive the hydrodynamic limit of the Kawasaki dynamics for the one-dimensional conservative system of unbounded real-valued spins with arbitrary strong, quadratic and finite-range interactions. This extends prior results for non-interacting spin systems. The result is obtained by adapting two scale approach of Grunewald, Otto, Villani and Westdickenberg combined with the authors' recent approach on conservative systems with strong interactions., Comment: 47 pages

The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough works in the mathematical understanding of ERGM, most notably in (Bhamidi, Bresler, Sly, 2011) through the analysis of a natural Heat-bath Glauber dynamics, and in (Chatterjee, Diaconis, 2013), (Eldan, Gross, 2018) via a large deviation theoretic perspective, several basic questions have remained unanswered owing to the lack of exact solvability unlike the much studied Curie-Weiss model (Ising model on the complete graph). In this paper, we establish a series of new concentration of measure results for the ERGM throughout the entire sub-critical phase, including a Poincar\'e inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the $W_1-$Wasserstein distance to Erd\H{o}s-R\'enyi graphs, previously obtained in (Reinert, Ross, 2017), is also presented. The arguments rely on translating temporal mixing properties of Glauber dynamics to static spatial mixing properties of the equilibrium measure and have the potential of being useful in proving similar functional inequalities for other Gibbsian systems beyond the perturbative regime., Comment: 35 pages. An argument in the previous version establishing exponential decay of entropy (Theorem 4) had an error and has been removed. To appear in Trans. Amer. Math. Soc

We develop a unified theory to analyze the microcanonical ensembles with several constraints given by unbounded observables. Several interesting phenomena that do not occur in the single constraint case can happen under the multiple constraints case. We systematically analyze the detailed structures of such microcanonical ensembles in two orthogonal directions using the theory of large deviations. First of all, we establish the equivalence of ensembles result, which exhibits an interesting phase transition phenomenon. Secondly, we study the localization and delocalization phenomena by obtaining large deviation results for the joint law of empirical distributions and the maximum component. Some concrete examples for which the theory applies will be given as well., Comment: 41 pages

We establish the well-posedness of stochastic differential equations possessing degenerate diffusions and singular drifts. We prove that SDEs defined on the homogeneous Carnot group, whose hypoelliptic diffusion part is given by the horizontal Brownian motion, admit a unique strong solution for a large class of singular drifts. It considerably generalizes the classical well-posedness results of singular SDEs with non-degenerate diffusions. It also provides an intermediate result between the Cauchy-Lipschitz theorem in ordinary differential equations and the result proved by Krylov and R\"ockner [38], which states the well-posedness of SDEs with the additive noise and singular drifts., Comment: 33 pages

We establish the well-posedness of SDE with the additive noise when a singular drift belongs to the critical spaces. We prove that if the drift belongs to the Orlicz-critical space $L^{q,1}([0,T],L^p_x)$ for $p,q\in (1,\infty)$ satisfying $\frac{2}{q}+\frac{d}{p} =1$, then the corresponding SDE admits a unique strong solution. We also derive the Sobolev regularity of a solution under the Orlicz-critical condition., Comment: 28 pages