Monday, 02 January 2023

We Will survey recent developments in invertibility of random matrices including breakthrough results of Tikhomirov on random matrices with i.i.d. Bernoulli entries and of Campos, Jenssen, Michelen, and Saharasbudhe on random symmetric matrices. This progress relies on a combination of probabilistic and geometric ideas.

We study the evolution of a random string in Poisson trap environment and compute the exponents of the limiting survival probability. It is well known that when the obstacles are hard then the problem is directly related to the sausage volume of the string. We shall briefly review the literature on particles moving in a trap environment, explain the key strategies that the particle may employ to survive and then discuss the same for the random string. This is joint work with Carl Mueller and Mathew Joseph

The metric associated with the Liouville quantum gravity (LQG) surface has been constructed through a series of recent works and several properties of its associated geodesics have been studied. In this talk we will discuss a proof of the folklore conjecture that the Euclidean Hausdorff dimension of LQG geodesics is strictly greater than 1 for all values of the so-called Liouville first passage percolation (LFPP) parameter ξ. This is deduced from an adaptation of a general criterion due to Aizenman and Burchard which in our case amounts to near-geometric bounds on the probabilities of certain crossing events for LQG geodesics in the number of crossings. We obtain such bounds using the axiomatic characterization of the LQG metric after proving a special regularity property for the Gaussian free field (GFF). If time permits, we will also discuss an analogous result for the LFPP geodesics.

We consider weakly confined particle systems in the plane, characterized by a large number of outliers away from a droplet, where the bulk of the particles accumulate in the many-particle limit. For Coulomb gases at determinantal inverse temperature, and zeros of random polynomials, we observe that the limiting outlier process only depends on the shape of the uncharged region containing them (and the global net excess charge).

In particular, for a determinantal Coulomb gas confined by a sufficiently regular background measure, the outliers in a simply connected uncharged region converge to the corresponding Bergman point process. Moreover, the outliers in different uncharged regions are asymptotically independent, even if the regions have common boundary points. The latter result is a demonstration of screening properties of the particle system.

Based on a joint work with R. Butez, D. García-Zelada, and A. Wennman.

We discuss two recent works on the motion of a tagged particle in 1D simple exclusion. One problem considers the Gumbel asymptotics of a lead particle in symmetric exclusion when starting from a step profile (with Michael Conroy, arXiv:2210.15550). Another problem finds optimal behaviors of a tagged particle in asymmetric exclusion subject to large deviation events (with SRS Varadhan, work-in-progress).

Tuesday, 03 January 2023

We Will survey recent developments in invertibility of random matrices including breakthrough results of Tikhomirov on random matrices with i.i.d. Bernoulli entries and of Campos, Jenssen, Michelen, and Saharasbudhe on random symmetric matrices. This progress relies on a combination of probabilistic and geometric ideas.

The Brunn-Minkowski theory in convex geometry concerns, among other things, the study of volumes, mixed volumes, and surface area measures of convex bodies. We discuss the generalizations of these concepts to Borel measures with density in $\R^n$, with particular focus on the standard Gaussian measure. After reviewing the weighted surface area and mixed measures (which are the weighted versions of mixed volumes), we introduce the mixed measures of three convex bodies, and explore inequalities related to all these quantities. The talk is based on joint work with Dylan Langharst, Matthieu Fradelizi, and Artem Zvavitch.

In this talk, we consider variants of the Boolean model based on Cox point processes, i.e., Poisson point processes in random environment. We establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points, also including their moments and in settings in which the environment features long-range correlations. We will also address rates of convergence of the percolation probability in a variety of asymptotic regimes and a sharp-threshold result for finite-range environments. If time allows, we finally present percolation results for a class of mobile device-to-device also in random environment. This is based on joint work with Christian Hirsch, András Tóbiás, Élie Cali, Stephen Muirhead and Alexander Hinsen.

Wednesday, 04 January 2023

We Will survey recent developments in invertibility of random matrices including breakthrough results of Tikhomirov on random matrices with i.i.d. Bernoulli entries and of Campos, Jenssen, Michelen, and Saharasbudhe on random symmetric matrices. This progress relies on a combination of probabilistic and geometric ideas.

The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analog of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behavior has remained out of reach. In this talk, based on joint work with Milind Hegde, we will discuss a geometric approach, closely connected to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach relies on a Gibbs resampling property of the KPZ equation as well as classical correlation inequalities, and yields a sharp understanding for a large class of initial data.

The hyperplane conjecture in convex geometry is a statement about the volume of a convex body and its hyperplane sections. Taking a measure-theoretic perspective to this problem, Bourgain highlighted the importance of the notion of a $\psi_2$-convex body, which captures integrability properties of linear images of the volume measure on the body. Despite this notion being introduced more than a quarter century ago, there are not many examples of such bodies. We describe several results on the $\psi_2$ (or more generally, $\psi_\alpha$) behavior of Schatten balls and their marginals. Along the way, we also establish some properties of the Haar measure on the orthogonal group that may be of independent interest. This is joint work with Grigoris Paouris.

It is well-known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko−Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas for the logarithmic energy of the mentioned models which can be of independent interest. Joint work with D. Chafaï et P. Youssef.

A celebrated problem in quantum mechanics concerns the effective mass of the Frohlich polaron in the strong coupling limit. Feynman gave a path integral formulation which relates it to a three dimensional Brownian motion with an attractive pair interaction of Kac type. The effective mass can be expressed as the inverse of the variance parameter of the self-interacting Brownian. There is a long standing conjecture about the asymptotic behavior of the effective mass for which Spohn gave a heuristic argument. A key property is that the interaction is shift invariant. Despite of considerable recent progress by Mukherjee, Varadhan, Spohn, Lieb, and Seiringer, the key problem is still open.

We present a much simpler model having a similar probabilistic structure which can be analyzed rigorously, and where the Spohn picture can be analyzed rigorously. No claim is made that the ""true"" polaron behaves in the same way.

This is joint work with Amir Dembo.

Thursday, 05 January 2023

Stochastic gradient descent (SGD) is one of the most, if not the most, influential optimization algorithms in use today.

It is the subject of extensive empirical and theoretical research, principally in justifying its performance at minimizing very large (high dimensional) nonlinear optimization. This talk is about the precise high—dimensional limit behavior (specifically generalization and training dynamics) of SGD in a high—dimensional least squares problems. High dimensionality is enforced by a family of resolvent conditions on the data matrix, and data-target pair, which can be viewed as a type of eigenvector delocalization. We show that the trajectory of SGD is quantitively close to the solution of a stochastic differential equation, which we call homogenized SGD, and whose behavior is explicitly solvable using renewal theory and the spectrum of the data.

Based on joint works with Courtney Paquette and Kiwon Lee (McGill), and Fabian Pedregosa, Jeffrey Pennington and Ben Adlam (Google Brain).

For various random matrix models of fairly general integral entries we discuss the probability that the cokernels are isomorphic to a given finite abelian group, or when they are cyclic. We will show that these statistics are asymptotically universal, given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra.

Based on joint works with R. V. Peski and M. M. Wood.

It is a standard fact that log-Sobolev inequality (LSI) implies modified log-Sobolev inequality (MLSI) which in turn implies Poincaré inequality. In this talk, we will consider the reverse implications on finite state spaces and prove using a regularization trick sharp comparisons between the LSI and MLSI constants. We present several applications of this: 1-Implementation of comparison procedures for MLSI constants of Markov chains; 2- Determining the LSI constant of Makov chains for which only MLSI is known; 3-Obtaining lower bounds on the MLSI constant of Markov chains for which LSI is determined. This allows us to answer a long-standing open problem on the mixing time of the switch chain on sparse regular bipartite graphs; obtain the first LSI for Zero-Range processes on arbitrary graphs; derive estimates on the MLSI constants for the Lamplighter chain on bounded degree graphs providing negative answers to two open problems of Montenegro-Tetali and Hermon-Peres respectively. This is based on joint works with Justin Salez and Konstantin Tikhomirov.

Logarithmically correlated fields (LCF) are random fields that exhibit a certain type of long range correlation. In the last two decades, they were shown to pop up in a variety of models, such as certain PDE's, Gaussian free fields, (planar) random walks, random matrices, random polynomials, polymers, and the Riemann zeta function. Various questions concerning the extremes of such fields have a common answer (some proven, some conjectured) in terms of extremes of Gaussian LCF. I will explain the common mechanism behind this universal phenomenon and its relation to branching random walks.

Friday, 06 January 2023

Monday, 09 January 2023

In this series of lectures, we will explore a medley of complex stochastic models and their applications to a range of problems from statistical physics to machine learning. These include continuous spin systems such as the Heisenberg and XY models, determinantal point processes and their generalizations, and the problem of learning under the action of latent group invariances. For continuous spin models, we will demonstrate a very general technique for obtaining Gaussian fluctuations at near-optimal rates, leveraging connections to the Lee-Yang theory of phase transitions and a quantitative generalization of the classical Marcinkiewicz Theorem. We will explore new kinds of determinantal processes arising from applications to the problems of stochastic gradient descent (SGD) and dimensionality reduction, and show how their stochastic geometry leads to connections with spiked models of random matrices and orthogonal polynomials. Finally, we will discuss the problem of learning under latent group actions, which is motivated by the well-known procedure of cryo-EM, and leads to an interesting interplay of structural constraints (such as sparsity) with harmonic analysis, group representations and invariant theory.

I will discuss some examples and some general limit theorems for interacting models defined on spatial random graphs constructed on stationary Euclidean point processes in a finite window. A prototypical example is co-operative sequential adsorption on the nearest neighbour graph built on the Ginibre point process. We establish the asymptotic normality, expectation, and variance asymptotics of statistics of the system evolution as the window size increases. I will try to indicate the general theory underlying these limit theorems. In particular, we will focus on the joint weak decay of correlations to be satisfied by the graph structure, point processes and initial states as well as sufficient conditions for the same. This is an on-going work with B. Blaszczyszyn (INRIA-ENS, Paris) and J. E. Yukich (Lehigh University, USA).

In this talk, I will present several results on the Anderson Hamiltonian with white noise potential in dimension 1. This operator formally writes « - Laplacian + white noise ». It arises as the scaling limit of various discrete models and its explicit potential allows for a detailed description of its spectrum. We will discuss localization of its eigenfunctions as well as the behavior of the local statistics of its eigenvalues. Around large energies, we will see that the eigenfunctions are localized and follow a universal shape given by the exponential of a Brownian motion plus a drift, a behavior already observed by Rifkind and Virag in tridiagonal matrix models. Based on joint works with Cyril Labbé.

We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. We prove that, for the walk started from a leaf vertex and stopped upon hitting the root, as the depth of the tree tends to infinity the maximal time spent at any leaf converges, under suitable scaling and centering, to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with square-root local-time process on the tree. Joint work with Marek Biskup (UCLA).

The Ising model is one of the most classical lattice models of statistical physics undergoing a phase transition. Initially imagined as a model for ferromagnetism, it revealed itself as a very rich mathematical object and a powerful theoretical tool to understand cooperative phenomena. Over one hundred years of its history, a profound understanding of its critical phase has been obtained. While integrability and mean-field behavior led to extraordinary breakthroughs in the two-dimensional and high-dimensional cases respectively, the model in three and four dimensions remained mysterious for years. In this talk, we will present recent progress in these dimensions based on a probabilistic interpretation of the Ising model relating it to percolation models.

Tuesday, 10 January 2023

I will present the solutions to two open problems about the Edwards--Anderson model of short-range spin glasses. First, I will show that the ground state is sensitive to small perturbations of the disorder, in the sense that a small amount of noise gives rise to a new ground state that is nearly orthogonal to the old one with respect to the site overlap inner product. Second, I will prove that one can overturn a macroscopic fraction of the spins in the ground state with an energy cost that is negligible compared to the size of the boundary of the overturned region --- a feature that is believed to be typical of spin glasses but clearly absent in ferromagnets. Together, these comprise the first mathematical proof of glassy behavior in a short-range spin glass model.

We will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the $λϕ^4$ fields over $R^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

In this series of lectures, we will explore a medley of complex stochastic models and their applications to a range of problems from statistical physics to machine learning. These include continuous spin systems such as the Heisenberg and XY models, determinantal point processes and their generalizations, and the problem of learning under the action of latent group invariances. For continuous spin models, we will demonstrate a very general technique for obtaining Gaussian fluctuations at near-optimal rates, leveraging connections to the Lee-Yang theory of phase transitions and a quantitative generalization of the classical Marcinkiewicz Theorem. We will explore new kinds of determinantal processes arising from applications to the problems of stochastic gradient descent (SGD) and dimensionality reduction, and show how their stochastic geometry leads to connections with spiked models of random matrices and orthogonal polynomials. Finally, we will discuss the problem of learning under latent group actions, which is motivated by the well-known procedure of cryo-EM, and leads to an interesting interplay of structural constraints (such as sparsity) with harmonic analysis, group representations and invariant theory.

I will describe the extremal landscape for the characteristic polynomial and orthogonal polynomials associated with the CbetaE ensemble. In particular, it will follow that the maximum on the unit circle of the (absolute value) of the log-determinant converges in distribution to the sum of two independent variables, one of which is Gumbel.

Joint work with Elliot Paquette.

Until the last few years, large deviation principles (LDP) for spectral statistics of random matrices were largely limited to the Gaussian ensembles. In a breakthrough work, Guionnet and Husson established the LDP for the largest eigenvalue of Wigner matrices having "sharp sub-Gaussian" entries with moment generating function bounded pointwise by that of the standard Gaussian; moreover, they established a universal rate function for the LDP (extending the GOE case). Their basic approach was to consider tilts of the Wigner matrix by spherical integrals, relating the upper tail for the largest eigenvalue to a difference of quenched and annealed free energies for a spherical spin glass model. Building on this work and followup work of Augeri–Guionnet–Husson, we obtain results for the general sub-Gaussian case by accounting for new localization phenomena that arise with heavier-than-Gaussian tails, which lead to non-universal rate functions. This is joint work with Raphael Ducatez and Alice Guionnet.

Wednesday, 11 January 2023

A nodal domain of a Laplacian eigenvector of a graph is a maximal connected subset of its vertices where it does not change sign. Sparse random regular graphs have been proposed as discrete toy models of ""quantum chaos"", and it has accordingly been conjectured by Y. Elon and experimentally observed by Dekel, Lee, and Linial that the high energy eigenvectors of such graphs have many nodal domains.

We prove superconstant (in fact, nearly linear in the number of vertices) lower bounds on the number of nodal domains of sparse random regular graphs, for sufficiently large Laplacian eigenvalues. The proof combines two different notions of eigenvector delocalization in random matrix theory as well as tools from graph limits and combinatorics. This is in contrast to what is known for dense Erdos-Renyi graphs, which have been shown to have only two nodal domains with high probability.

Joint work with Shirshendu Ganguly, Theo McKenzie, and Sidhanth Mohanty.

We will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the $λϕ^4$ fields over $R^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

In this series of lectures, we will explore a medley of complex stochastic models and their applications to a range of problems from statistical physics to machine learning. These include continuous spin systems such as the Heisenberg and XY models, determinantal point processes and their generalizations, and the problem of learning under the action of latent group invariances. For continuous spin models, we will demonstrate a very general technique for obtaining Gaussian fluctuations at near-optimal rates, leveraging connections to the Lee-Yang theory of phase transitions and a quantitative generalization of the classical Marcinkiewicz Theorem. We will explore new kinds of determinantal processes arising from applications to the problems of stochastic gradient descent (SGD) and dimensionality reduction, and show how their stochastic geometry leads to connections with spiked models of random matrices and orthogonal polynomials. Finally, we will discuss the problem of learning under latent group actions, which is motivated by the well-known procedure of cryo-EM, and leads to an interesting interplay of structural constraints (such as sparsity) with harmonic analysis, group representations and invariant theory.

It has been an active research topic to understand the spectral statistics of random matrices. An important and interesting question is investigating large deviation properties of spectral observables, such as the empirical spectral measure and extreme eigenvalues. In the absence of exact formulas such as in Gaussian ensembles, these questions are quite hard to analyze. In particular, for sparse random matrices, even investigating the typical behavior of eigenvalues is nontrivial. In this talk, we will survey recent progress in understanding spectral large deviations of such sparse random networks. Based on joint works with Shirshendu Ganguly and Ella Hiesmayr.

In the regime where the mean degree is at least logarithmic, the edge eigenvalues of a sparse Wigner matrix sticks to the edges of the support of the semicircle law. We show that in this sparsity regime, the large deviations of the largest eigenvalue of a sparse Wigner matrix with sub-Gaussian entries are dominated by either the emergence of a high degree vertex with a high weight or that of a clique with high weights. Interestingly, the rate function obtained is discontinuous at the typical value of the largest eigenvalue, which accounts for the fact that its large deviation behaviour is generated by finite rank perturbations. This complements the results of Ganguly-Nam and Ganguly-Hiesmayr-Nam which settle the case where the mean degree is constant. This is a joint work with Anirban Basak

Thursday, 12 January 2023

We will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the $λϕ^4$ fields over $R^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

We will discuss the connection between the universality problem for random polynomials and the anti-concentration problem. We shall survey several methods to attack the latter and new approaches that are used in recent papers to study large classes of random polynomials, including random orthogonal polynomials. This is based on several joint works with Yen Do, Doron Lubinsky, Hoi Nguyen, Igor Pritsker, and Van Vu.

The statics and dynamics of mean-field models of spin glasses have been studied in-depth by the physics community since the '70s. At the heart of this is the trade-off between the notions of replica symmetry breaking, shattering, and metastability. I will survey the current mathematical understanding of these ideas in the “simple” case of the spherical p-spin model. I will start by recalling how the landscape complexity can be used to understand of the “replica symmetry breaking” phase following the work of Auffinger–Ben Arous–Cerny and Subag. I'll then turn to our recent joint work with Ben Arous on the “replica symmetric” phase. Here we prove the existence of a shattering phase and show that metastable states exist up to an even higher temperature as predicted by Barrat–Burioni–Mezard. This latter work is based on a Thouless–Anderson–Palmer decomposition which builds on the ideas of Subag. I will end by presenting a series of open questions and conjectures surrounding sharp phase boundaries for shattering and metastability.

This talk will touch on joint work with: A. Auffinger (Northwestern), G. Ben Arous (Courant), R. Gheissari (Northwestern), and I. Tobasco (UIC)

The theory of qauntum chaos aims to discribe quantum mechanical states in an environment where the classical dynamic is chaotic. The guiding example is the Laplace-Beltrami operator on a compact hyperbolic smooth manifold and it is conjectured by Rudnick and Sarnack that the underlying chaotic classical dynamics on such manifolds reults in delocalization properties of the eigenfunctions of the Laplace-Beltrami operator.

In this talk we shall consider a toy model for this: We will show how Lagrangian states propagated by the semi-groupe induced by a suitable random Schrödinger operator converge locally to a stationary monochromatic istropic Gaussian field.

This is joint work with M. Ingremeau.

Friday, 13 January 2023