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.

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.