ICTS

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

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 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.

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. 

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 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.

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.

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).

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. 

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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 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.

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.