On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization

TL;DR

This paper establishes a connection between contractive couplings, spectral independence, and entropy decay in Markov chains, leading to improved bounds on mixing times and log-Sobolev constants for various spin systems.

Contribution

It introduces new techniques linking contractive couplings to spectral independence and entropy factorization, enabling optimal bounds for mixing times and entropy decay in spin systems.

Findings

Proves that contractive couplings imply spectral independence.

Establishes entropy factorization from spectral independence.

Derives optimal mixing time bounds for Glauber and Swendsen-Wang dynamics.

Abstract

For general spin systems, we prove that a contractive coupling for any local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics, arbitrary heat-bath block dynamics, and the Swendsen-Wang dynamics. This reveals a novel connection between probabilistic techniques for bounding the convergence to stationarity and analytic tools for analyzing the decay of relative entropy. As a corollary of our general results, we obtain $O(n\log{n})$ mixing time and $\Omega(1/n)$ modified log-Sobolev constant of the Glauber dynamics for sampling random $q$-colorings of an $n$-vertex graph with constant maximum degree $\Delta$ when $q > (11/6 - \epsilon_0)\Delta$ for some fixed $\epsilon_0>0$. We also obtain $O(\log{n})$ mixing time and $\Omega(1)$ modified log-Sobolev constant of the Swendsen-Wang dynamics for the ferromagnetic Ising model on an $n$-vertex graph of constant maximum degree when the parameters of the system lie in the tree uniqueness region. At the heart of our results are new techniques for establishing spectral independence of the spin system and block factorization of the relative entropy. On one hand we prove that a contractive coupling of a local Markov chain implies spectral independence of the Gibbs distribution. On the other hand we show that spectral independence implies factorization of entropy for arbitrary blocks, establishing optimal bounds on the modified log-Sobolev constant of the corresponding block dynamics.