Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

TL;DR

This paper introduces a new lower bound on the spectral gap of Glauber dynamics for spectrally independent spin systems, enabling improved mixing time results and sampling algorithms for graph colorings, hardcore models, and matchings.

Contribution

It provides a novel spectral gap lower bound that extends the regime of known results, improving mixing time bounds and sampling algorithms for various graph models.

Findings

Spectral gap bounds for triangle-free graphs with proper colorings.

Efficient approximation of the hardcore model partition function for graphs with girth at least 6.

Fast sampling of matchings in random graphs with improved runtime.

Abstract

We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent $q$-spin system on a graph $G = (V,E)$ with maximum degree $\Delta$. Notably, for several interesting examples, our bound covers the entire regime of $\Delta$ excluded by arguments based on coupling with the stationary distribution. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments: (1) We show that for a triangle-free graph $G = (V,E)$ with maximum degree $\Delta \geq 3$, the Glauber dynamics for the uniform distribution on proper $k$-colorings with $k \geq (1.763\dots + \delta)\Delta$ colors has spectral gap $\tilde{\Omega}_{\delta}(|V|^{-1})$. Previously, such a result was known either if the girth of $G$ is at least $5$ [Dyer et.~al, FOCS 2004], or under restrictions on $\Delta$ [Chen et.~al, STOC 2021; Hayes-Vigoda, FOCS 2003]. (2) We show that for a regular graph $G = (V,E)$ with degree $\Delta \geq 3$ and girth at least $6$, and for any $\varepsilon, \delta > 0$, the partition function of the hardcore model with fugacity $\lambda \leq (1-\delta)\lambda_{c}(\Delta)$ may be approximated within a $(1+\varepsilon)$-multiplicative factor in time $\tilde{O}_{\delta}(n^{2}\varepsilon^{-2})$. Previously, such a result was known if the girth is at least $7$ [Efthymiou et.~al, SICOMP 2019]. (3) We show for the binomial random graph $G(n,d/n)$ with $d = O(1)$, with high probability, an approximately uniformly random matching may be sampled in time $O_{d}(n^{2+o(1)})$. This improves the corresponding running time of $\tilde{O}_{d}(n^{3})$ due to [Jerrum-Sinclair, SICOMP 1989; Jerrum, 2003].