Probability Seminar

Back to Probability Group

• When: Thursdays at 2:30 pm
• Where: 901 Van Vleck Hall
• Organizers: Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
• To join the probability seminar mailing list: email probsem+subscribe@g-groups.wisc.edu.
• To subscribe seminar lunch announcements: email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu

Past Seminars

Spring 2025

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

January 23, 2025:

No seminar

January 30, 2025: Promit Ghosal (UChicago)

Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond

Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near-optimal rates with polynomial dimensional dependence and extend these results to bilinear-enhanced kernels.

February 6, 2025: Subhabrata Sen (Harvard)

Community detection on multi-view networks

The community detection problem seeks to recover a latent clustering of vertices from an observed random graph. This problem has attracted significant attention across probability, statistics and computer science, and the fundamental thresholds for community recovery have been characterized in the last decade. Modern applications typically collect more fine-grained information on the units under study. For example, one might measure relations of multiple types among the units, or observe an evolving network over time. In this talk, we will discuss the community detection problem on such ‘multi-view’ networks. We will present some new results on the fundamental thresholds for community detection in these models. Finally, we will introduce algorithms for community detection based on Approximate Message Passing.

This is based on joint work with Xiaodong Yang and Buyu Lin (Harvard University).

February 13, 2025: Hanbaek Lyu (UW-Madison)

Large random matrices with given margins

We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization,  Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such a margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the typical table. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two potential functions, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$.  We also propose a generalized Sinkhorn algorithm for computing typical tables and establish its linear convergence. We derive several new results for random contingency tables from our general framework.

Based on a joint work with Sumit Mukherjee (Columbia)

February 20, 2025: Mustafa Alper Gunes (Princeton)

Characteristic Polynomials of Random Matrices, Exchangeable Arrays & Painlevé Equations

Joint moments of characteristic polynomials of unitary random matrices and their derivatives have gained attention over the last 25 years, partly due to their conjectured relation to the Riemann zeta function. In this talk, we will consider the asymptotics of these moments in the most general setting allowing for derivatives of arbitrary order, generalising previous work that considered only the first derivative. Along the way, we will examine how exchangeable arrays and integrable systems play a crucial role in understanding the statistics of a class of infinite Hermitian random matrices. Based on joint work with Assiotis, Keating and Wei.

February 27, 2025: Souvik Dhara (Purdue)

Propagation of Shocks on Networks: Can Local Information Predict Survival?

Abstract: Complex systems are often fragile, where minor disruptions can cascade into dramatic collapses. Epidemics serve as a prime example of this phenomenon, while the 2008 financial crisis highlights how a domino effect, originating from the small subprime mortgage sector, can trigger global repercussions. The mathematical theory underlying these phenomena is both elegant and foundational, profoundly shaping the field of Network Science since its inception. In this talk, I will present a unifying mathematical model for network fragility and cascading dynamics, and explore its deep connections to the theory of local-weak convergence, pioneered by Benjamini-Schramm and Aldous-Steele.

March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy)

What conditional probability could (probably) be

According to orthodox probability theory, when B has probability zero, the conditional probability of A given B can depend on the partition or sub-sigma-field that B is relativized to. This relativization to sub-sigma-fields, a hallmark of Kolmogorov's theory of conditional expectation, is traditionally seen as appropriate in a treatment of conditioning with continuous variables, and it is what allows the theory to preserve Total Disintegrability, a generalization of the Law of Total Probability to uncountable partitions. In this talk, I will argue that although the relativization of conditional probability to sub-sigma-fields has advantages, it also has an underrecognized cost: it leads to puzzles for the treatment of iterated conditioning. I will discuss these puzzles and some possible implications for the foundations of conditional probability.

This talk is based on joint work with Snow Zhang (UC Berkeley).

March 13, 2025: Klara Courteaut (Courant)

The Coulomb gas on a Jordan arc

We study a Coulomb gas on a sufficiently smooth simple 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 a quantity involving the partition function of the log-gas on [−1,1] and the Fredholm determinant of the arc-Grunsky operator. Alternatively, we can express this quantity in terms of the Loewner energy of a specific Jordan curve associated with the arc. 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 asymptotic variance given by the Dirichlet energy of the test function.

Based on joint work with Kurt Johansson and Fredrik Viklund.

March 20, 2025: Ewain Gwynne (UChicago)

Random walk reflected off of infinity

Let $\mathcal G$ be an infinite graph --- not necessarily one-ended --- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\mathcal G$ which reaches $\infty$ in finite time and ``reflects off of $\infty$'' infinitely many times.

We show that the Aldous-Broder algorithm for the random walk reflected off of $\infty$ gives the free uniform spanning forest (FUSF) on $\mathcal G$. Furthermore, Wilson's algorithm for the random walk reflected off of $\infty$ gives the FUSF on $\mathcal G$ on the event that the FUSF is connected, but not in general.

We also apply the theory of random walk reflected off of $\infty$ to study random planar maps in the universality class of supercritical Liouville quantum gravity (LQG), equivalently LQG with central charge $c \in (1,25)$. Such random planar maps are infinite, with uncountably many ends. We define a version of the Tutte embedding for such maps under which they conjecturally converge to LQG. We also conjecture that the free uniform spanning forest on these maps is connected when $c > 16$ (but not when $c < 16$); and that there is an infinite open cluster for critical percolation on these maps when $c < 95/4$ (but not when $c > 95/4$).

Based on joint work with Jinwoo Sung.

March 27, 2025: SPRING BREAK

No seminar

April 3, 2025: Jimme He (OSU)

Random growth models with half space geometry

Abstract: Random growth models in 1+1 dimension capture the behavior of interfaces evolving in the presence of noise. These models are expected to exhibit universal behavior, but we are still far from proving such results even in relatively simple models. A key development which has led to recent progress is the discovery of exact formulas for certain models with rich algebraic structure, leading to asymptotic results. I will discuss work on the asymmetric simple exclusion process with one open boundary, as well as applications to rates of convergence for a Markov chain.

April 10, 2025: Evan Sorensen (Columbia)

Viscous shock fluctuations in KPZ

I will discuss a recent preprint with Alex Dunlap, where we study ``V-shaped" solutions to the KPZ equation. These are solutions having asymptotic slopes \theta > 0 and -\theta at plus and minus infinity, respectively. We show that there are no V-shaped invariant measures for the KPZ equation, which, combined with recent work of Janjigian, Rassoul-Agha, and Seppalainen, completes the classification of the extremal invariant measures for the KPZ equation. To accomplish this, we study the fluctuations of viscous shocks in the KPZ equation under some special choices of initial data. While V-shaped invariant measures in a fixed frame of reference do not exist, we give an explicit description of a family of V-shaped invariant measures from the perspective of a shock.  

April 17, 2025:

No seminar

April 24, 2025: William Leeb (University of Minnesota, Twin Cities)

Signal recovery in the high-noise, high-dimensional regime

This talk will describe recent work on recovering high-dimensional signals corrupted by high levels of noise. The first part of the talk will explain the connection between the Wiener filter, singular value shrinkage, and Stein's method for covariance estimation, and review optimal shrinkage in the spiked covariance model. We will then present extensions to heteroscedastic noise and linearly-corrupted observations. Time permitting, we will also give an overview of the related class of orbit recovery problems.

May 1, 2025: Hai-Xiao Wang (UCSD)

Singular values of sparse random rectangular matrices: emergence of outliers at criticality

Consider the random bipartite Erdos-Renyi graph $G(n, m, p)$, where each edge with one vertex in $V_{1}=[n]$ and the other vertex in $V_{2}=[m]$ is connected with probability $p$ with $n \geq m$. For the centered and normalized adjacency matrix $H$, it is well known that the empirical spectral measure will converge to the Mar\v{c}enko-Pastur (MP) distribution. However, this does not imply that the largest (resp. smallest) singular values will converge to the right (resp. left) edge, especially in the sparse case when $p = o(1)$. In Dumitriu and Zhu 2024, it was proved that when $np = \omega(\log(n))$, there are almost surely no outliers outside the compact support of the MP law. In this paper, we consider the critical sparsity regime $np =O(\log(n))$, where we denote $p = b\log(n)/\sqrt{mn}$ for some constant $b>0$, with constant aspect ratio $\ratio = n/m \geq 1$. For the first time in the literature, we quantitatively characterize the emergence of outlier singular values, as follows. For explicit $b_{\star}$ and $b^{\star}$ functions of the aspect ratio $\ratio$, we prove that when $b > b_{\star}$, there is no outlier outside the bulk; when $b^{\star}< b < b_{\star}$, outlier singular values are present only outside the right edge of the MP law; and when $b < b^{\star}$, outliers are present on both sides---all with high probability. Moreover, the locations of those outliers are precisely characterized by a function depending on the largest and smallest degree vertices of the sampled random graph. Our results follow the path forged by Alt, Ducatez and Knowles 2021, and can be extended to sparse random critical matrices with bounded entries.