From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk

TL;DR

This paper links coupling methods to spectral independence to establish rapid mixing of Glauber dynamics for spin systems, providing new bounds and concentration results for proper list-colorings on bounded-degree graphs.

Contribution

It introduces a novel approach connecting couplings to spectral independence, enabling optimal mixing bounds and concentration results for Glauber dynamics in list-colorings.

Findings

Established asymptotically optimal lower bounds on log-Sobolev constants.

Proved $O(n\,log n)$ mixing time for proper list-colorings with specific list sizes.

Derived Chernoff-type concentration bounds for Hamming Lipschitz functions.

Abstract

We show that the existence of a "good"' coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More specifically, we only require the expected distance between successive iterates under the coupling to be summable, as opposed to being one-step contractive in the worst case. Combined with recent local-to-global arguments \cite{CLV21}, we establish asymptotically optimal lower bounds on the standard and modified log-Sobolev constants for the Glauber dynamics for sampling from spin systems on bounded-degree graphs when a curvature condition \cite{Oll09} is satisfied. To achieve this, we use Stein's method for Markov chains \cite{BN19, RR19} to show that a "good" coupling for a local Markov chain yields strong bounds on the spectral independence of the distribution in the sense of \cite{ALO20}. Our primary application is to sampling proper list-colorings on bounded-degree graphs. In particular, combining the coupling for the flip dynamics given by \cite{Vig00, CDMPP19} with our techniques, we show optimal $O(n\log n)$ mixing for the Glauber dynamics for sampling proper list-colorings on any bounded-degree graph with maximum degree $\Delta$ whenever the size of the color lists are at least $\left(\frac{11}{6} - \epsilon\right)\Delta$, where $\epsilon \approx 10^{-5}$ is small constant. While $O(n^{2})$ mixing was already known before, our approach additionally yields Chernoff-type concentration bounds for Hamming Lipschitz functions in this regime, which was not known before. Our approach is markedly different from prior works establishing spectral independence for spin systems using spatial mixing \cite{ALO20, CLV20, CGSV20, FGYZ20}, which crucially is still open in this regime for proper list-colorings.