Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chains (extended 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 on some space $\Omega$, 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. In particular, we: (i) Give the first O(n log n) bound for mixing time of the hardcore-model (of arbitrary degree) in the tree-uniqueness regime, under Glauber dynamics, (ii) Give the first optimal mixing bounds for Ising models in the uniqueness regime under any external fields, (iii) Prove a KL-divergence decay bound for log-concave sampling via the Restricted Gaussian Oracle, which achieves optimal mixing under any exp (n)-ivarm start, (iv) Prove a logarithmic-Sobolev inequality for near-critical Ferromagnetic Ising models, recovering in a simple way a variant of a recent result by Bauerschmidt and Dagallier.

I. Introduction

Suppose that we would like to sample from a measure v on some set . For the sake of discussion, suppose that either is the Boolean hypercube or common algorithm is to find a Markov chain whose stationary distribution is v and which exhibits good mixing bounds.