Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chains

Understand

Two recent and seemingly-unrelated techniques for proving mixing bounds for Markov chains are: (i) the framework of Spectral Independence, introduced by Anari, Liu and Oveis Gharan, and its numerous extensions, which have given rise to several breakthroughs in the analysis of mixing times of discrete Markov chains and (ii) the Stochastic Localization technique which has proven useful in establishing mixing and expansion bounds for both log-concave measures and for measures on the discrete hypercube.

• In this paper, we introduce a framework which connects ideas from both techniques.
• Our framework unifies, simplifies and extends those two techniques.
• In its center is the concept of a localization scheme which, to every probability measure, assigns a martingale of probability measures which localize in space as time evolves.