I am organizing the UC San Diego Probability Seminar in 2023-2024. This is a hybrid seminar, and if you are interested in joining (or giving a talk) just send me an email. Here is a list of talks and abstracts.
Title: Nonlinear spiked covariance matrices and signal propagation in neural network models
Abstract: Abstract: In this talk, we will discuss recent work on the extreme eigenvalues of the sample covariance matrix with a spiked population covariance. Extending previous random matrix theory, we will characterize the spiked eigenvalues outside the bulk distribution and their corresponding eigenvectors for a nonlinear version of the spiked covariance model. Our result shows the universality of the spiked covariance model with the same quantitative spectral properties as a linear spiked covariance model. In the proof, we will present a deterministic equivalent for the Stieltjes transform for any spectral argument separated from the support of the limit spectral measure. Then, we will apply this new result to deep neural network models. We will describe how spiked eigenstructure in the input data propagates through the hidden layers of a neural network with random weights. As a second application, we can study a simple regime where the weight matrix has a rank-one signal component over gradient descent training and characterize the alignment of the target function. This is a joint work with Denny Wu and Zhou Fan.
Title: Equilibrium in functional stochastic games with mean-field interaction
Abstract: We study a general class of finite-player stochastic games with mean-field interaction where the linear-quadratic cost functional includes linear operators acting on controls in L^2. We propose a new approach for deriving the Nash equilibrium of these games in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations which can be solved. Moreover, by deriving stability results for the system of Fredholm equations, we obtain the convergence of the finite-player Nash equilibrium to the mean-field equilibrium in the infinite player limit. Our general framework includes examples of stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.This is joint work with Eduardo Abi Jaber (Ecole Polytechnique) and Eyal Neuman (Imperial College London). The paper is available at https://ssrn.com/abstract=4470883.
Title: The critical percolation probability is local
Abstract: Around 2008, Schramm conjectured that the critical percolation probability $p_c$ of a transitive graph is entirely determined by the local geometry of the graph, subject to the global constraint that $p_c <1$. Previous works had verified the conjecture in various special cases, including nonamenable graphs of high girth (Benjamini, Nachmias and Peres 2012); Cayley graphs of abelian groups (Martineau and Tassion 2013); nonunimodular graphs (Hutchcroft 2017 and 2018); graphs of uniform exponential growth (Hutchcroft 2018); and graphs of (automatically uniform) polynomial growth (Contreras, Martineau and Tassion 2022). In this talk I will describe joint work with Hutchcroft (https://arxiv.org/abs/2310.10983) in which we resolve this conjecture.
Title: Coupling method and the fundamental gap problem on the sphere.
Abstract: The reflection coupling method on Riemannian manifolds is a powerful tool in the study of harmonic functions and elliptic operators. In this talk, we will provide an overview of some fundamental ideas in stochastic analysis and the coupling method. We will then focus on applying these ideas to the study of the fundamental gap problem on the sphere. Based on joint work with Gang Yang and Guofang Wei.
Abstract: Random interface growth is all around us: tumors, bacterial colonies, infections, and propagating flame fronts. The KPZ equation is a stochastic PDE central to a class of random growth phenomena. In this talk, I will explain how to combine tools from probability, partial differential equations, and integrable systems to understand the behavior of the KPZ equation when it exhibits unusual growth.
Title: Spectral Gap Estimates for Mixed $p$-Spin Models at High Temperature
Abstract: We consider general mixed $p$-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the $N$-spin system to that of suitably conditioned subsystems.
Title: Stochastic Embeddings of Hyperbolic Metric Spaces
Abstract: This talk is based on ongoing work of the speaker. We will discuss the stochastic embeddability of snowflakes of finite Nagata-dimensional spaces into ultrametric spaces and the induced stochastic embeddings of their hyperbolic fillings into trees. Several results follow as applications, for example: (1) For any uniformly concave gauge $\omega$, the Wasserstein 1-metric over $([0,1]^n,\omega(\|\cdot\|))$ biLipschitzly embeds into $\ell^1$. (2) The Wasserstein 1-metric over any finitely generated Gromov hyperbolic group biLipschitzly embeds into $\ell^1$.
Marco Carfagnini 