Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses

TL;DR

This paper demonstrates the universality of spectral independence in spin systems, showing that a relaxation time of O(n) implies spectral independence, and applies this to establish optimal mixing times and concentration results for certain distributions.

Contribution

The paper proves that spectral independence is a universal property linked to relaxation times and applies this to analyze Glauber dynamics for a broad class of distributions, including high-temperature spin glasses.

Findings

Spectral independence implies O(n) relaxation time.

Glauber dynamics mixes in O(n) for distributions with smooth Hamiltonians.

Results enable improved concentration and learning guarantees.

Abstract

We study Glauber dynamics for sampling from discrete distributions $\mu$ on the hypercube $\{\pm 1\}^n$. Recently, techniques based on spectral independence have successfully yielded optimal $O(n)$ relaxation times for a host of different distributions $\mu$. We show that spectral independence is universal: a relaxation time of $O(n)$ implies spectral independence. We then study a notion of tractability for $\mu$, defined in terms of smoothness of the multilinear extension of its Hamiltonian -- $\log \mu$ -- over $[-1,+1]^n$. We show that Glauber dynamics has relaxation time $O(n)$ for such $\mu$, and using the universality of spectral independence, we conclude that these distributions are also fractionally log-concave and consequently satisfy modified log-Sobolev inequalities. We sharpen our estimates and obtain approximate tensorization of entropy and the optimal $\widetilde{O}(n)$ mixing time for random Hamiltonians, i.e. the classically studied mixed $p$-spin model at sufficiently high temperature. These results have significant downstream consequences for concentration of measure, statistical testing, and learning.