I organized the UC San Diego Probability Seminar in 2023-2024. Here is a list of talks and abstracts. All seminars are at 11am PST in AP&M 6402.
Date: June 6 2024
Speaker: Lutz Warnke (UCSD)
Title: PDEs in random graph theory: Analyticity of scaling limits
Abstract: We consider a generalization of the maximum degree in random graphs. Given a rooted tree T , let Xv denote the number of copies of T rooted at v in the binomial random graph Gn,p . We ask the question where the maximum Mn=max{X1,...,Xn} of Xv over all vertices is concentrated. For edge-probabilities p=p(n) tending to zero not too fast, the maximum is asymptotically attained by the vertex of maximum degree. However, for smaller edge probabilities p=p(n) , the behavior is more complicated: our large deviation type optimization arguments reveal that the behavior of Mn changes as we vary p=p(n) , due to different mechanisms that can make the maximum large.
Date: May 30 2024
Speaker: David Weisbart (UCR)
Title: p-Adic Brownian Motion is a Scaling Limit
Abstract: The Laplace operator is the infinitesimal generator of Brownian motion with a real state space. The Vladimirov operator, a $p$-adic analogue of the Laplace operator, similarly gives rise to Brownian motion with a $p$-adic state space. This talk aims to introduce the concept of a $p$-adic Brownian motion and demonstrate a further similarity with its real analogue: $p$-adic Brownian motion is a scaling limit of a discrete-time random walk on a discrete group. Attendees need not have prior knowledge of $p$-adic analysis, as the talk will provide a brief review of necessary background information.
Date: May 16 2024
Speaker: Garret Tresch (Texas A&M)
Title: Stochastic Embeddings of Graphs into Trees
Abstract: As the shortest path metric on a weighted tree can be embedded isometrically into a finite ℓ1 space, a Lipschitz embedding of a given graph into ℓ1 can be obtained by constructing a low distortion embedding into a tree. Conversely, while there are various topological properties of graphs that guarantee controlled distortion Lipschitz embeddings into ℓ1 (k-outerplanar, series-parallel, low Euler characteristic), it is still often the case that such a graph embeds quite poorly into a tree. By introducing the notion of a stochastic embedding into a family of trees one can find more general concrete embeddings into ℓ1 then those limited by a single tree. In fact, it is known that every graph with n vertices embeds stochastically into trees with distortion O(log(n)). Nevertheless, this upper bound is sharp for graphs such as expanders, grids and, by a recent joint work with Schlumprecht, a large class of "fractal-like" series-parallel graphs called slash powers. In this talk we introduce an equivalent characterization of stochastic distortion called expected distortion and after proving a mild extension of a result of Gupta regarding poor tree embeddings of a cycle, inductively lower bound the expected distortion of generalized Laakso graphs found in most nontrivial slash power families.
Date: May 9 2024
Speaker: John Peca-Medlin (University of Arizona, Tucson)
Title: Random permutations using GEPP
Abstract: Gaussian elimination with partial pivoting (GEPP) remains the most used dense linear solver. For an x n matrix A, GEPP results in the factorization PA = LU whereL and U are lower and upper triangular matrices and P is a permutation matrix. If A is a random matrix, then the associated permutation from the P factor is random. When is this a uniform permutation? How many cycles are in its disjoint cycle decomposition (which equivalently answers how many GEPP pivot movements are needed on A)? What is the length of the longest increasing subsequence of this permutation? We will provide some statistical answers to these questions for select random matrix ensembles and transformations. For particular butterfly permutations, we will present full distributional descriptions for these particular statistics. Moreover, we introduce a random butterfly matrix ensemble that induces the Haar measure on a full 2-Sylow subgroup of the symmetric group on a set of size 2ⁿ.
Date: May 2 2024
Speaker: Zichao Wang (UCSD)
Title: Nonlinear spiked covariance matrices and signal propagation in neural network models
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.
Date: April 25 2024
Speaker: Konstantinos Panagiotou (Ludwig Maximilian University of Munich)
Title: Limit Laws for Critical Dispersion on Complete Graphs
Abstract: click here.
Date: April 11 2024
Speaker: Moritz Voss (UCLA)
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).
Date: March 21 2024
Speaker: Philip Easo (Caltech)
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 in which we resolve this conjecture.
Date: February 08 2024
Speaker: Karl-Theodor Sturm (Bonn)
Title: Wasserstein Diffusion on Multidimensional Spaces
Abstract: Given any closed Riemannian manifold M, we construct a reversible diffusion process on the space P(M) of probability measures on M, that is, - reversible w.r.t. the entropic measure Pβ on P(M), heuristically given as dPβ (μ) = 1/Z e^(- β Ent(μ | m)) dP0(μ). - associated with a regular Dirichlet form with Carré Du Champ derived from the Wasserstein gradient in the sense of Otto calculus E_W(f)=\liminf_{g \to f}\ 1/2 int_{(P)(M)} | \nabla_W g |^2(μ) dPβ (μ); - non-degenerate, at least in the case of the n-sphere and the $-torus.
Date: November 16 2023
Speaker: Gunhee Chow (UCSB)
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.
Date: November 2 2023
Speaker: Yier Lin (The University of Chicago)
Title: The atypical growth in a random interface
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.
Date: October 26 2023
Speaker: Arka Adhikari (Stanford)
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.
Date: October 5 2023
Speaker: Chris Gartland (UCSD)
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 $\ell^1$.