From independent walks to interacting particles

Abstract: Suppose that labelled particles occupy the vertices of a graph. Whenever an edge rings, the particles at its endpoints exchange places. The resulting process, known as the interchange process, can equivalently be viewed as a random walk on the symmetric group.

How does this interaction affect the time it takes the system to approach equilibrium? I will describe new results comparing the interchange process with independent random walks, resolving a conjecture of Oliveira and establishing cutoff—an abrupt transition to equilibrium, for both systems around the same time.

A theme of the talk is that this enormous many-particle system can sometimes be understood through the motion and collisions of just a few particles, leading to connections between probability, analysis and the representation theory of the symmetric group.

***** There will be coffee, tea and snacks at PIMS from 2:30 PM to 3 PM.

***** UBC Math Colloquium 2026-2027 Zoom info:

Meeting ID: 663 8015 7380

Passcode: 31415