Regularized modified log-Sobolev inequalities, and comparison of Markov chains

Abstract: In this work, we develop a comparison procedure for the Modified log-Sobolev Inequality (MLSI) constants of two reversible Markov chains on a finite state space. As an application, we provide a sharp estimate of the MLSI constant of the switch chain on the set of simple bipartite regular graphs of size n with a fixed degree d. Our estimate implies that the total variation mixing time of the switch chain is of order O(n log(n)). The result is optimal up to a multiple depending on d and resolves an old open problem. Based on joint work with Pierre Youssef.

Bio:
Konstantin Tikhomirov is an Assistant Professor at the School of Mathematics at the Georgia Institute of Technology, with interests in discrete probability, combinatorics, convex geometry, and applications to data analysis.