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

TL;DR

This paper introduces a unified framework connecting Spectral and Stochastic Localization techniques to analyze Markov chain mixing times, simplifying proofs and deriving new bounds such as an $O(n \, \log n)$ mixing time for Glauber dynamics.

Contribution

It unifies two recent techniques into a single framework using localization schemes, simplifying analysis and extending results in Markov chain mixing time bounds.

Findings

Unified framework for Spectral and Stochastic Localization techniques.

Simplified proofs of existing mixing bounds.

First $O(n \log n)$ mixing time bound for Glauber dynamics in certain models.

Abstract

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. As it turns out, to every such scheme corresponds a Markov chain, and many chains of interest appear naturally in this framework. This viewpoint provides tools for deriving mixing bounds for the dynamics through the analysis of the corresponding localization process. Generalizations of concepts of Spectral Independence and Entropic Independence naturally arise from our definitions, and in particular we recover the main theorems in the spectral and entropic independence frameworks via simple martingale arguments (completely bypassing the need to use the theory of high-dimensional expanders). We demonstrate the strength of our proposed machinery by giving short and (arguably) simpler proofs to many mixing bounds in the recent literature, including giving the first $O(n \log n)$ bound for the mixing time of Glauber dynamics on the hardcore-model (of arbitrary degree) in the tree-uniqueness regime.