Time: Wednesdays from 1:30-2:30pm (EST), unless otherwise noted.

Some talks may be in-person, but the default will be Zoom talks. In-person talks will be in Recitation 121 and will be noted on the schedule.

Abstracts

Wednesday, August 25. Yun Li University of Wisconsin - Madison(Zoom talk)
Limits of the circular Jacobi beta-ensemble

We study the point process limits of the circular Jacobi beta-ensemble and the real orthogonal beta-ensemble. Using the differential operator framework introduced by Valkó and Virág, we identify the point process scaling limits as spectra of certain stochastic differential operators. This framework also allows us to prove the convergence of the normalized characteristic polynomials of the finite models to certain random analytic functions. In this talk, I will first review the theoretic operator framework and these constructions. Then I will present the operator level convergence and several characterizations of the limiting objects. This talk is based on joint work with Benedek Valkó.

Wednesday, September 8. Pierre-Yves Gaudreau Lamarre University of Chicago(In-person talk)
Number rigidity in the spectrum of random Schrödinger operators

In this talk, I will discuss recent progress in the understanding of the structure in the spectrum of random Schrödinger operators. More specifically, I will introduce the concept of number rigidity in point processes and discuss recent efforts to understand its occurrence in the spectrum of random Schrödinger operators. Based on joint works with Promit Ghosal (MIT), Wenxuan Li (UChicago), and Yuchen Liao (Warwick).

Wednesday, September 15. Jim Fill The Johns Hopkins University, Department of Applied Mathematics and Statistics
(Zoom talk)
Breaking Multivariate Records

For general dimension \(d\), we identify, with proof, the asymptotic conditional distribution of the number of (Pareto) records broken by an observation given that the observation sets a record.
Fix \(d\), and let \({\mathcal K}(d)\) be a random variable with this distribution. We show that the (right) tail of \({\mathcal K}(d)\) satisfies
\[
{\mathbb P}({\mathcal K}(d) \geq k) \leq \exp\left[ - \Omega\!\left( k^{(d - 1) / (d^2 - 2)} \right) \right]\mbox{ as $k \to \infty$}
\]
and
\[
{\mathbb P}({\mathcal K}(d) \geq k) \geq \exp\left[ - O\!\left( k^{1 / (d - 1)} \right) \right]\mbox{ as \(k \to \infty\)}.
\]
When \(d = 2\), the description of \({\mathcal K}(2)\) in terms of a Poisson process agrees with the main result from
Fill [Comb. Probab. Comput. 30 (2021) 105--123] that the distribution of \({\mathcal K}(2)\) is Geometric\((1/2)\) with support \(\{0, 1, \ldots\}\).
We show that \({\mathbb P}({\mathcal K}(d) \geq 1) = \exp[-\Theta(d)]\) as \(d \to \infty\); in particular, \({\mathcal K}(d) \to 0\) in probability as \(d \to \infty\).

Wednesday, September 22. Yiran Liu Purdue University(In-person talk)
Diffusion Approximations for Cox/\(G_t\)/\(\infty\) in Random
Environment

We study Cox/\(G_t\)/\(\infty\) queues driven by Cox processes in a fast oscillatory
random environment. Within the framework of stochastic homogenization, we
can establish diffusion approximations to the rescaled number-in-system process
by proving functional central limit theorems (FCLTs). This framework allows
us to obtain both quenched and annealed limits. At the quantitative level, we
identify subcritical and supercritical regimes which indicate the relative domi-
nance between the randomness in the arrival intensity and that in the service
times. In this talk, I will discuss the recent results in our analysis of Cox/\(G_t\)/\(\infty\) queues. Based on joint work with Harsha Honnappa, Samy Tindel, and Aaron
Yip.

Wednesday, September 29. Otávio Menezes Purdue University
TBD