A sharp log-Sobolev inequality for the multislice

Abstract

We determine the log-Sobolev constant of the multi-urn Bernoulli–Laplace diffusion model with arbitrary parameters, up to a small universal multiplicative constant. Our result extends a classical estimate of Lee and Yau (1998) and confirms a conjecture of Filmus, O’Donnell and Wu (2018). Among other applications, we completely quantify the small-set expansion phenomenon on the multislice, and obtain sharp mixing-time estimates for the colored exclusion process on various graphs.