BIMSA >
Probability and Dynamical Systems Seminar
Probability and Dynamical Systems Seminar
Homogenization of weakly reinforced Pólya urns on countable networks
Homogenization of weakly reinforced Pólya urns on countable networks
Organizers
Speaker
Time
Tuesday, November 3, 2026 3:15 PM - 4:15 PM
Venue
A3-3-301
Online
Zoom 482 240 1589
(BIMSA)
Abstract
We study a system of interacting Pólya urns on a countable graph of uniformly bounded degree. Each vertex carries an independent Poisson clock. When it rings, an incident edge is selected with probability proportional to the $\alpha$-th power of its current weight, and that weight is increased by one. For $0\leq\alpha<1$ and positive, uniformly bounded clock rates, we prove that the edge weights divided by time converge almost surely, at every edge, to a deterministic positive profile independent of the initial weights. This resolves the homogenization conjecture of Couzinié and Hirsch.
The main theme of the talk is the connection between this stochastic evolution and an infinite-dimensional dynamical system. I will explain how logarithmic time reveals the associated ODE, and how limits of time-shifted trajectories lead naturally to solutions defined for all real times. A Liouville theorem then rules out nonconstant limiting trajectories. The talk will emphasize the model, its background, and the main ideas behind the proof. No familiarity with stochastic approximation will be assumed.
The main theme of the talk is the connection between this stochastic evolution and an infinite-dimensional dynamical system. I will explain how logarithmic time reveals the associated ODE, and how limits of time-shifted trajectories lead naturally to solutions defined for all real times. A Liouville theorem then rules out nonconstant limiting trajectories. The talk will emphasize the model, its background, and the main ideas behind the proof. No familiarity with stochastic approximation will be assumed.
Speaker Intro
Shuo Qin has been the first Chern Instructor at BIMSA. He obtained a Ph.D. in mathematics in 2024 from New York University under the supervision of Prof. Pierre Tarrès. His work is in probability theory, especially in random processes with memory or reinforcement.