Spectral independence of compact groups

Abstract: Let $G_1$ and $G_2$ be compact simple (real or $p$-adic) Lie groups, and let $\mu_1$ and $\mu_2$ be symmetric probability measures on $G_1$ and $G_2$. Under mild conditions on $\mu_1$ and $\mu_2$, the distribution of $\mu_i$ random walks on $G_i$ converges to the uniform measure, and the speed of convergence is governed by the spectral gap. A coupling of $\mu_1$ and $\mu_2$ is any probability measure $\mu$ on $G_1 \times G_2$ whose marginal distributions are $\mu_1$ and $\mu_2$, respectively . A natural question is under what conditions a spectral gap for all couplings depending on spectral gaps of $\mu_1$ and $\mu_2$ can be established. In this talk, I will present results in this direction which are based on joint work with Alireza S. Golsefidy and Amir Mohammadi.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar