Probability Seminar

Mean-field games (MFGs) model non-cooperative games among large populations of agents and are widely applied in areas such as traffic flow, finance, and epidemic control. Inverse mean-field games address the challenge of inferring environmental factors from observed agent behavior. The coupled forward-backward structure of MFG equations makes solving these problems difficult and adds even greater complexity to their inverse problems. In this talk, I will introduce a policy iteration method for solving inverse MFGs. This method simplifies the problem by decoupling it into solving linear PDEs and linear inverse problems, leading to significant computational efficiency. The approach is flexible, accommodating a variety of numerical methods and machine learning tools. I will also present theoretical results that guarantee the convergence of our proposed method, along with numerical examples demonstrating its accuracy and efficiency.

Langevin algorithms, integral to Markov Chain Monte Carlo methods, are crucial in machine learning, particularly for Bayesian inference in high-dimensional models and addressing challenges in stochastic non-convex optimization prevalent in deep learning. This talk delves into the practical aspects of stochastic Langevin algorithms through three illuminating examples. First, it explores their role in non-convex optimization, focusing on their efficacy in navigating complex landscapes. The discussion then extends to decentralized Langevin algorithms, emphasizing their relevance in distributed optimization scenarios, where data is dispersed across multiple sources. Lastly, the focus shifts to constrained sampling, aiming to sample from a target distribution subject to constraints. In each scenario, we introduce new algorithms with convergence guarantees and showcase their performance and scalability to large datasets through numerical examples.

Let w be a word in a free group. A few years ago, Magee and I, relying on a work of Calegari, discovered that the stable commutator length of w, which is a well-studied topological invariant, can also be defined in terms of certain Fourier coefficients of w-random unitary matrices. But there are very natural ways to tweak the random-matrix side of this story: one may consider, for example, w-random permutations or w-random orthogonal matrices, and apply the same definition to obtain other "stable" invariants of w. Are these invariants interesting? Do they have, too, alternative topological/combinatorial definitions? In a joint work with Yotam Shomroni, we present concrete conjectures and begin to answer some of them. No background is assumed — I will define all notions.

The expression "Parisi formula" refers to a variational formula postulated by Parisi in 1980 to give the limiting free energy of the Sherrington–Kirkpatrick (SK) spin glass.  The SK model was originally conceived as a mean-field description for disordered magnetism, and has since become a mathematical prototype for frustrated disordered systems and high-complexity functions.  In recent years, there has been an effort to extend the Parisi framework to various generalizations of the SK model, raising new physical questions met with fresh mathematical challenges.  In this talk, I will share some developments in this evolving story.  Based on joint works with Leila Sloman and Youngtak Sohn.

We study a Coulomb gas on a sufficiently smooth Jordan arc in the complex plane, at arbitrary positive temperature. We show that, as the number of particles tends to infinity, the partition function converges to an expression involving the partition function of the gas on [−1,1], a power of the capacity of the curve, and the Fredholm determinant of the arc-Grunsky operator. We also obtain an asymptotic formula for the Laplace transform of linear statistics for sufficiently regular test functions. This shows that the centered empirical measure converges to a Gaussian field with explicit asymptotic mean and variance given by the Dirichlet energy of the test function. Based on joint work with Kurt Johansson and Fredrik Viklund.

For d>= 1, it is an easy fact that a graph G must have minimum degree at least d in order to be rigid in R^d. In the talk I will present a recent work, joint with A. Lew, E. Nevo and Y. Peled, where we show that the threshold probability for a random graph G(n, p) to be d-rigid, coincides with the known threshold for having minimum degree d. This extends the classical proof for a random graph to be connected, which corresponds to the case d=1.

A fundamental feature of turbulence, first predicted by Richardson, is the explosive dispersion of advected particles: at sufficiently late times, the separation of initially close particles grows independently of their initial separation. This implies that turbulent flows exhibit spontaneous stochasticity, or the non-uniqueness of Lagrangian trajectories in the zero-noise limit. Spontaneous stochasticity is mathematically equivalent to another well-documented phenomenon in turbulence: the anomalous dissipation of passive scalars. However, a rigorous mathematical demonstration of these effects in fluid models remains elusive. To investigate anomalous dissipation in a more tractable setting, one seeks to construct incompressible vector fields that explicitly exhibit this behaviour. Only recently have explicit deterministic vector fields with anomalous dissipation been constructed.

In this talk, I will provide an overview of spontaneous stochasticity/anomalous dissipation and discuss my recent work with Keefer Rowan, where we construct a universal total anomalous dissipator—a vector field that completely dissipates any initial data in unit time in the vanishing noise limit. Specifically, we construct a vector field such that the laws of the associated SDEs remain diffuse even as the noise vanishes. In fact, as the noise approaches zero, the laws converge to the uniform distribution on the torus.

We describe how one can obtain the pair correlation function of the Sine-beta process for beta=2n using the random operator framework. Our method recovers the classical formulas for beta=2 and 4. For general beta=2n, we identify the pair correlation function in terms of an ODE system, and also in terms of a matrix-valued generalization of hypergeometric functions.

Joint with Yahui Qu (UW–Madison).

Branching Brownian motion describes a growing swarm of particles that move and multiply stochastically. In multiple dimensions, the frontier of this population is governed by an associated "derivative martingale," which converges to a random field on the sphere. In this talk, we will explore the irregularity of this limiting field: it is almost surely discontinuous almost everywhere in d≥3.

Much of the recent rigorous progress on the classical Ising model was driven by a new detailed understanding of its stochastic geometric representations. The extent of couplings between the Ising model and its FK, random current and high-temperature representations is ever-increasing. A framework unifying the relations will be presented, along with some surprising implications including:

• percolation of the uniform even graph is not monotone in the domain;
• percolation of the high-temperature expansion and single current is not monotone in the parameter;
• absence of exponential decay for the random current corresponding to a supercritical Ising model.

Building upon the dynamic programming principle for set-valued functions arising from many applications, we will present a new notion of set-valued PDEs. The key component is a set-valued Ito formula, characterizing the flows on the surface of the dynamic sets. In the context of multivariate control problems, we establish the wellposedness of the set-valued HJB equations, which extends the standard HJB equations in the scalar case to the multivariate case. As an application, we discuss moving scalarization, constructed using the classical solution of the set-valued HJB equation. Additionally, we introduce the concept of set values for games under Nash equilibrium, along with the corresponding PDE, and explore its geometric properties. The talk is based on joint work with Jianfeng Zhang and ongoing work joint with Nizar Touzi and Jianfeng Zhang.

Given a collection of points in a normed space, the corresponding ``geometric graph" is obtained by connecting any pair of points with distance less than one. Say that a graph $G$ is "geometrically embeddable" into a normed space $X$ if there exist points in $X$ whose geometric graph is isomorphic to $G$. Geometric embeddability arises naturally at the intersection of combinatorics, metric geometry, and data science.

While criteria for geometric embeddings are well-studied in Euclidean space, essentially nothing is known outside this setting. We address this gap. Our result is that asymptotically almost every regular graph $G$ on $N$ vertices has the following ``universal" non-embedding property: there is no normed space of dimension less than $c \log N$ admitting a geometric embedding of $G$. This is sharp. The proof is based on an efficient multiscale ``seeded" epsilon--net argument.

(joint with Konstantin Tikhomirov; arxiv.org/abs/2501.09142)

I will introduce basic concepts such as the concentration and fluctuation of Φ^4 Gibbs type measures from the perspectives of statistical physics, quantum field theory, and probability theory. The focus will be on the low temperature behavior and the thermodynamic limit of these probability measures, with particular attention to fluctuations around the soliton manifold.

We study dynamic finite-player and mean-field stochastic games within the framework of Markov perfect equilibria (MPE). Unlike their continuous-time analogues, discrete-time finite-player games generally do not admit unique MPE. However, we show that uniqueness is remarkably recovered when the time steps are sufficiently small, and we provide examples demonstrating the necessity of this assumption. This result, established without relying on any monotonicity conditions, underscores the importance of inertia in dynamic games. Furthermore, we discuss different learning algorithms and prove their convergence to the unique MPE.

The Coulomb gas is a statistical physics model consisting of N particles interacting with electrostatic repulsion and with a confining potential. I will first review results on the microscopic structure on the gas. Then, I will show how a certain subharmonic structure associated with the k-point correlation function arises. This structure implies new bounds on quantities such as the furthest particle from the origin while generalizing bounds known for the Ginibre ensemble, and it also explains how Poisson point process statistics take over in the high-temperature regime.

Suppose that a continuous-time, stochastic diffusion (i.e., the Susceptible-Infected process) spreads on an unknown graph. We only observe the time at which the diffusion reaches each vertex, i.e., the set of infection times. What can be learned about the unknown graph from the infection times? While there is far too little information to learn individual edges in the graph, we show that certain high-level properties — such as the number of vertices of sufficiently high degree, or super-spreaders — can surprisingly be determined with certainty. To achieve this goal, we develop a suite of algorithms that can efficiently detect vertices of degree asymptotically greater than $\sqrt{n}$ from infection times, for a natural and general class of graphs with n vertices. To complement these results, we show that our algorithms are information-theoretically optimal: there exist graphs for which it is impossible to tell whether vertices of degree larger than $n^{1/2 - \epsilon}$ exist from vertices' infection times, for any $\epsilon > 0$. Finally, we discuss the broader implications of our ideas for change-point detection in non-stationary point processes. This talk is based on joint work with Anna Brandenberger (MIT) and Elchanan Mossel (MIT).

In 2004, motivated by connections of random matrix theory to number theory, Diaconis and Gamburd showed a fascinating connection between the enumeration problem of magic squares (squares filled with integers with row and column sum constraints) and the moments of the ‘secular coefficients’ of random matrices, when the size of the matrix tends to infinity. These are the coefficients in the monomial expansion of a characteristic polynomial, or equivalently, the elementary symmetric polynomials of the eigenvalues of this random matrix. It turns out that this characteristic polynomial has a limit, when the matrix size tends to infinity. It converges to a random fractal, the holomorphic multiplicative chaos. We describe this process on the unit circle, and show how it can be connected even more strongly to random matrices, and how magic square combinatorics are a type of ‘signature’ of this holomorphic multiplicative chaos. We’ll review some open questions about these objects, and discuss some links between this and other stochastic processes such as the Gaussian multiplicative chaos, the circular beta-ensemble and random multiplicative function.