Spectral Independence via Stability and Applications to Holant-Type Problems
Zongchen Chen, Kuikui Liu, Eric Vigoda
TL;DR
The paper builds a general, nearly black-box bridge from zero-free regions of multivariate partition functions to spectral independence of Gibbs distributions, enabling near-optimal $O(n\log n)$ mixing for Glauber dynamics on bounded-degree graphs across Holant-type models. By proving that stability underpins rapid mixing, it unifies zero-free region techniques with MCMC analysis and yields fast sampling/counting algorithms that surpass Barvinok-style polynomial interpolation bounds in this regime. The main contributions include new theorems translating $\Gamma$-stability (and its variants) into explicit spectral-independence constants, preservation of stability under pinnings, and comprehensive application to binary symmetric Holant problems, weighted graph homomorphisms, and tensor-network contractions. The results significantly broaden the scope of zero-free region methods, offering practical, scalable samplers for edge covers, even subgraphs (Ising with field), line-graph spin models, and other Holant-type structures on bounded-degree graphs, with potential impact on related combinatorial counting tasks and statistical physics models.
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 $λ$ then spectral independence holds at $λ$. 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.