Probability Seminar

Back to Probability Group

• When: Thursdays at 2:30 pm
• Where: 901 Van Vleck Hall
• Organizers: Hongchang Ji, Ander Aguirre, Hai-Xiao Wang
• 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

Fall 2025

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

We usually end for questions at 3:20 PM.

September 11, 2025: David Renfrew (Binghamton U.)

Singularities in the spectrum of random block matrices

We consider the density of states of structured Hermitian and non-Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.

September 18, 2025: JE Paguyo (McMaster U.)

Asymptotic behavior of the hierarchical Pitman-Yor and Dirichlet processes

The Pitman-Yor process is a discrete random measure specified by a concentration parameter, discount parameter, and base distribution, and is used as a fundamental prior in Bayesian nonparametrics. The hierarchical Pitman-Yor process (HPYP) is a generalization obtained by randomizing the base distribution through a draw from another Pitman-Yor process. It is motivated by the study of groups of clustered data, where the group specific Pitman-Yor processes are linked through an intergroup Pitman-Yor process. Setting both discount parameters to zero recovers the celebrated hierarchical Dirichlet process (HDP), first introduced by Teh et al. In this talk, we discuss our recent work on the asymptotic behavior of the HPYP and HDP. First, we establish limit theorems associated with the power sum symmetric polynomials for the vector of weights of the HDP as the concentration parameters tend to infinity. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics. Second, we consider a random sample of size $N$ from a population whose type distribution is given by the vector of weights of the HPYP and study the large $N$ asymptotic behavior of the number of clusters in the sample. Our approach relies on a random sample size representation of the number of clusters through the corresponding non-hierarchical process. This talk is based on joint work with Stefano Favaro and Shui Feng.

September 25, 2025: Chris Janjigian (Purdue U.)

Boundaries of random walks in random potentials

This talk will discuss various notions of boundaries at infinity of random walks in random potentials. Recent results on existence and uniqueness will be presented for a class of models that generalizes first- and last-passage percolation, random walks in random environments, and directed polymers. The resulting boundary structures are related to jointly stationary distributions, geodesic rays, Busemann functions, harmonic functions and the associated Martin boundary, and extremal Gibbs-DLR measures.

Based primarily on joint work with Sean Groathouse and Firas Rassoul-Agha.

October 2, 2025: Elliot Paquette (McGill U.)

From magic squares, through random matrices, and to the multiplicative chaos

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 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.

October 16, 2025: Zachary Selk (Florida State U.)

On the Onsager-Machlup Function for the \Phi^4 Measure

The \Phi^4 measure is a measure arising in effective quantum field theory as arguably the simplest example of a nontrivial QFT, modelling the self-interaction of a single scalar quantum field. This measure can be constructed through a procedure known as stochastic quantization. Stochastic quantization seeks to construct a measure on an infinite dimensional space with a given Gibbs-type ``density function" as the invariant measure of a stochastic PDE, in analogy with Langevin dynamics of stochastic ODEs. Both the \Phi^4 measure and its associated stochastic quantization PDE involve nonlinearities of distributions, necessitating renormalization procedures via tools like Wick calculus, regularity structures or paracontrolled calculus. Although the \Phi^4 measure has been constructed in dimensions 1,2 and 3, the question of whether these measures have the desired ``density function" remains open. Although in infinite dimensions, density functions are typically thought to not exist as there is no reference Lebesgue measure, there is a notion of a probability density function that extends to infinite dimensions called the Onsager-Machlup (OM) functional. One pathology of OM theory is that different metrics can lead to different OM functionals, or OM functionals can fail to exist under reasonable metrics. In a joint work with Ioannis Gasteratos (TU Berlin), we study the OM functional for the \Phi^4 measure. In dimension 1, the OM functional is what is desired under naive choices of metrics. In dimension 2, the OM functional is what is desired if we choose a metric analogous to the rough paths metric. In dimension 3, naive approaches don't work and the situation is complicated.

October 30, 2025: Ander Aguirre (UW-Madison)

Edgeworth expansion and random polynomials

In this talk, we discuss the large n limit of the number of real zeros of random Weyl polynomials with arbitrary non-Gaussian coefficients ($N_{n, \xi}$). Random polynomial ensembles often exhibit features of both universality and non-universality. For instance, in the trigonometric ensemble, the variance is linear in $n$ the degree of the polynomial $P_n(x)$, a signature of lack of correlation among sufficiently far apart roots. This phenomenon is universal in that it suffices to assume that the coefficients $\xi$ have bounded moments. However, the exact multiplicative constant depends on the first few moments of $\xi$. Our main result states that for the Weyl ensemble the expectation scales as $\mathbb{E} N_{n, \xi}=\frac{2}{\pi} \sqrt{n} +C_{\xi}+o(1)$ where we identify the exact non-universal $C_{\xi}$. Similarly, for the variance we establish the scaling $\operatorname{var} N_{n, \xi}=\operatorname{var} N_{n, G}+o(\sqrt{n})$. Our result crucially relies on an Edgeworth expansion for random walks in $\R^2$ and $\R^4$ arising from the Weyl polynomials. This enables the application of the Kac-Rice formula to study the expectation and variance of the number of real roots. We also discuss the role of the arithmetic structure of the Weyl coefficients in providing concentration probability estimates. Joint work with Hoi Nguyen and Jingheng Wang.