Homogenization is the approximation of a complex, “disordered” system by a simpler, “ordered” one. Picture a walker on a grid. In each step, the walker chooses to walk along a neighboring edge with equal probability. At large scales, the walker approximates Brownian motion. But what if some edges are more likely to be traversed than others? I will discuss recent advances in the theory of quantitative homogenization which make it possible to analyze random walk with drift and other models in probability. Joint work with Scott Armstrong and Tuomo Kuusi.

We study the Brownian evolution of large non-Hermitian matrices and show that their log-determinant converges to a 2+1-dimensional Gaussian field in the Edwards-Wilkinson regularity class, i.e. logarithmically correlated for the parabolic distance. This gives a dynamical extension of the celebrated result by Rider and Virag (2006) proving that the fluctuations of the eigenvalues of Gaussian non-Hermitian matrices converge to a 2-dimensional log-correlated field. Our result, previously not known even in the Gaussian case, holds out of equilibrium for general matrices with i.i.d. entries. We also study the extremal values of these fields and demonstrate their logarithmic dependence on the matrix dimension.

The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. We give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, corresponding to a lower bound on the geometric entanglement of random quantum states. The proof is based on spin-glass methods, the Kac—Rice formula, and recent progress coming from random matrices. Joint work with Stéphane Dartois.

Approximate message passing (AMP) has emerged as a powerful framework for the design and analysis of iterative algorithms for high dimensional inference problems involving regression and low-rank matrix factorization. The basic form of an AMP algorithm consists of a recursion defined on a random matrix. Under suitable conditions, the distribution of this recursion can be well approximated by a Gaussian process whose mean and covariance are defined via a recursive process called state evolution. This talk will briefly summarize some of the key ideas in AMP (no background is assumed). I will then describe a new approach for analyzing these algorithms that constructs an explicit coupling between the AMP iterates and a Gaussian process, Under mild regularity conditions, this coupling argument provides simple and interpretable guarantees on the non-asymptotic behavior of AMP algorithms. Related work can be found in the arXiv papers: https://arxiv.org/abs/2405.08225 and https://arxiv.org/abs/2306.15580

The six-vertex model, also known as the square-ice model, is one of the central and most studied systems of 2d statistical mechanics. It offers various combinatorial interpretations. One of them involves molecules of water on the square grid; another one deals with non-intersecting lattice paths, which can be also viewed as level lines of an integer-valued height function. Despite many efforts since the 1960s, the limit shapes for the height function are still unknown in general situations. However, we recently found ways to compute them in a degeneration, which leads to a low density of corners of paths (or, equivalently, of horizontal/vertical molecules of water). I will report on the progress in this direction emphasizing various unusual features: appearance of hyperbolic PDEs; discontinuities in densities; connections to random permutations.

We study the half-space KPZ equation with a Neumann boundary condition, starting from stationary Brownian initial data. We derive a variance identity that links the fluctuations of the height function to the transversal fluctuations of a half-space polymer model. We then establish optimal fluctuation exponents for the height function in both the subcritical and critical regimes, along with corresponding estimates for the polymer endpoint. Based on a joint work with Yu Gu.

Random regular graphs form a ubiquitous model for chaotic systems. However, the spectral properties of their adjacency matrices have proven difficult to analyze because of the strong dependence between different entries. In this talk, I will describe recent work that shows that despite this, the fluctuation of eigenvalues of the adjacency matrix are of the same order as for Gaussian matrices. This gives an optimal error term for Friedman's theorem that the second eigenvalue of the adjacency matrix of a random regular graph converges to the spectral radius of an infinite regular tree. Crucial is tight analysis of the Green’s function of the adjacency operator and an analysis of the change of the Green's function after a random edge switch. This is based on joint work with Jiaoyang Huang and Horng-Tzer Yau.

The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing controlled multi-agent systems. The study of such systems has seen increased interest in recent years, due to their ubiquity in applications coming from macroeconomics, social behavior, and telecommunications. When the number of agents becomes large, the model can be formally replaced by one involving a mean-field description of the population, analogously to similar models in statistical physics. Justifying this continuum limit is often nontrivial and is sensitive to the type of stochastic noise influencing the population, i.e. idiosyncratic or systemic. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces. In particular, we develop new stability and regularity results for the equations. These allow for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type. We conclude with a discussion of some further problems for which the techniques for equations on Wasserstein space may be amenable.

One way to understand the concentration of the norm of a random matrix X with Gaussian entries is to apply a standard concentration inequality, such as the one for Lipschitz functions of i.i.d. standard Gaussian variables, which yields subgaussian tail bounds on the norm of X. However, as was shown by Tracy and Widom in 1990s, when the entries of X are i.i.d. the norm of X exhibits even sharper concentration. The phenomenon of a function of many i.i.d. variables having strictly smaller tails than those predicted by classical concentration inequalities is sometimes referred to as «superconcentration», a term originally dubbed by Chatterjee. I will discuss novel results that can be interpreted as superconcentration inequalities for the norm of X, where X is a Gaussian random matrix with independent entries and an arbitrary variance profile. We can also view our results as a nonhomogeneous extension of Tracy-Widom-type upper tail estimates for the norm of X.

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model has a one parameter family {p(a), a > 0} of product form mutually singular stationary distributions. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure p(a) if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to p(a) weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of p(a) for each a. Results on extremality and ergodicity of p(a) will be presented. Finally, I will describe some recent results on fluctuations of the Atlas model from inhomogeneous stationary profiles. This is based on joint work with Sayan Banerjee and Peter Rudzis.

The random distribution Holomorphic multiplicative chaos (HMC) with Gaussian inputs is recently introduced independently by Najnudel, Paquette, and Simm as a limiting object on the unit complex circle of characteristic polynomial of circular beta ensembles, and by Soundararajan and Zaman as an analogue of random multiplicative functions. In this talk, we will explore this rich connection between HMC and random matrix theory, number theory, and Gaussian multiplicative chaos. We will also discuss the regularity of this distribution, alongside the fractional moments and tightness of its Fourier coefficients (also referred to as secular coefficients). Furthermore, we introduce non-Gaussian HMC, and discuss the Gaussian universality and two phase transitions phenomenon in the fractional moments of its secular coefficients. A transition from global to local effect is observed, alongside an analysis of the critical local-global case. As a result, we unveil the regularity of some non-Gaussian HMC and tightness of their secular coefficients. Based on joint work with Zhenyuan Zhang.

Recently, Markov-chain Monte Carlo methods based on non-reversible piecewise deterministic Markov processes (PDMP) are under growing attention, thanks to the increase in performance they usually bring. Beyond their numerical efficacy, the non-reversible and piecewise deterministic characteristics of these processes prompt interesting questions, regarding for instance ergodicity proof and convergence bounds. During this talk, I will particularly focus on the obtained results and open problems left while considering PDMP evolution of particle systems, both in an equilibrium and out-of-equilibrium setting. Hardcore particle systems have embodied a testbed of choice since the first implementations of Markov chain Monte Carlo in the 50’s. Even today, the entropic barriers they exhibit are still resisting to the state-of-the-art MCMC sampling methods. During this talk, I will review the recent developments regarding sampling such systems and discuss the dynamical bottlenecks that are yet to be solved.

Directed polymers are a statistical mechanics model for random growth. Their partition functions are solutions to a discrete stochastic heat equation. This talk will discuss the logarithmic derivatives of the partition functions, which are solutions to a discrete stochastic Burgers equation. Of interest is the success or failure of the “one force-one solution principle” for this equation. I will reframe this question in the language of polymers, and share some surprising results that follow. Based on joint work with Louis Fan and Timo Seppäläinen.