Wednesday, 21 November 2018
Keivan Mallahi-Karai: Spectral independence and random walks on products of compact simple groups
Let \(G_1\) and \(G_2\) be either compact simple groups or finite simple groups of Lie type. Let \(\mu_1\) and \(\mu_2\) be symmetric probability measures on \(G_1\) and \(G_2\), respectively. Under some mild conditions on \(\mu_1\), \(\mu_2\), one knows that the distribution of the random walk on \(G_i\) driven by \(\mu_i\) converges to the uniform distribution, and the speed of convergence is governed by the spectral gap of \(\mu_i\). A coupling of \(\mu_1\) and \(\mu_2\) is a probability measure \(\mu\) on \(G_1 \times G_2\) with marginal distributions \(\mu_1\) and \(\mu_2\), respectively. Under what conditions does \(\mu\) have a spectral gap depending on the gaps of \(\mu_1\) and \(\mu_2\)?In this talk I will first review some of the old and new methods for establishing spectral gaps, mainly based on pioneering works of Kazhdan and Bourgain-Gamburd and then discuss the question stated in the previous paragraph. This talk is based on a joint work with Amir Mohammadi and Alireza Salehi-Golsefidy.