Spectral Independence Beyond Uniqueness using the topological method

TL;DR

This paper introduces a topological method to analyze spectral independence in Gibbs distributions, leading to improved bounds for rapid mixing of Glauber dynamics on graphs with spectral radius constraints.

Contribution

It develops a novel topological approach to bound eigenvalues in spectral independence, extending analysis beyond the uniqueness regime for Gibbs distributions.

Findings

Provides new bounds for spectral independence using the spectral radius of adjacency and non-backtracking matrices.

Derives improved rapid mixing bounds for Glauber dynamics on graphs with spectral radius less than maximum degree.

Introduces techniques connecting influence matrices to topological structures like self-avoiding walks.

Abstract

We present novel results for fast mixing of Glauber dynamics using the newly introduced and powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. In our results, the parameters of the Gibbs distribution are expressed in terms of the spectral radius of the adjacency matrix of $G$, or that of the Hashimoto non-backtracking matrix. The analysis relies on new techniques that we introduce to bound the maximum eigenvalue of the pairwise influence matrix $\mathcal{I}^{\Lambda,\tau}_{G}$ for the two spin Gibbs distribution $\mu$. There is a common framework that underlies these techniques which we call the topological method. The idea is to systematically exploit the well-known connections between $\mathcal{I}^{\Lambda,\tau}_{G}$ and the topological construction called tree of self-avoiding walks. Our approach is novel and gives new insights to the problem of establishing spectral independence for Gibbs distributions. More importantly, it allows us to derive new -- improved -- rapid mixing bounds for Glauber dynamics on distributions such as the Hard-core model and the Ising model for graphs that the spectral radius is smaller than the maximum degree.