Spectral Independence via Stability and Applications to Holant-Type Problems

TL;DR

This paper establishes a connection between polynomial stability and MCMC convergence, leading to optimal sampling algorithms for Holant problems and spin systems on bounded-degree graphs, with improved running times over previous methods.

Contribution

It introduces a novel link between polynomial stability and spectral independence, enabling faster mixing time bounds for Markov chains in Holant problems and spin models.

Findings

Achieves $O(n\log n)$ mixing times for Glauber dynamics on bounded-degree graphs.

Provides improved sampling algorithms for weighted edge covers and ferromagnetic Ising models.

Extends applications to various models including graph homomorphisms and tensor networks.

Abstract

This paper formalizes connections between stability of polynomials and convergence rates of Markov Chain Monte Carlo (MCMC) algorithms. We prove that if a (multivariate) partition function is nonzero in a region around a real point $\lambda$ then spectral independence holds at $\lambda$. As a consequence, for Holant-type problems (e.g., spin systems) on bounded-degree graphs, we obtain optimal $O(n\log n)$ mixing time bounds for the single-site update Markov chain known as the Glauber dynamics. Our result significantly improves the running time guarantees obtained via the polynomial interpolation method of Barvinok (2017), refined by Patel and Regts (2017). There are a variety of applications of our results. In this paper, we focus on Holant-type (i.e., edge-coloring) problems, including weighted edge covers and weighted even subgraphs. For the weighted edge cover problem (and several natural generalizations) we obtain an $O(n\log{n})$ sampling algorithm on bounded-degree graphs. The even subgraphs problem corresponds to the high-temperature expansion of the ferromagnetic Ising model. We obtain an $O(n\log{n})$ sampling algorithm for the ferromagnetic Ising model with a nonzero external field on bounded-degree graphs, which improves upon the classical result of Jerrum and Sinclair (1993) for this class of graphs. We obtain further applications to antiferromagnetic two-spin models on line graphs, weighted graph homomorphisms, tensor networks, and more.