Optimal mixing for two-state anti-ferromagnetic spin systems

TL;DR

This paper establishes optimal bounds on the mixing time of Glauber dynamics for anti-ferromagnetic two-spin systems in the tree uniqueness regime, demonstrating rapid convergence under certain spectral conditions.

Contribution

It proves the first optimal lower bound on the modified log-Sobolev constant for these systems, leading to tight mixing time bounds in the uniqueness regime.

Findings

Optimal $ ilde{O}(n \, \log n)$ mixing time for anti-ferromagnetic two-spin systems.

MLS constant lower bound of $ ilde{\Omega}(n^{-1})$ in the tree uniqueness regime.

Results hold for systems with arbitrary maximum degree, depending only on the spectral gap.

Abstract

We prove an optimal $\Omega(n^{-1})$ lower bound for modified log-Sobolev (MLS) constant of the Glauber dynamics for anti-ferromagnetic two-spin systems with $n$ vertices in the tree uniqueness regime. Specifically, this optimal MLS bound holds for the following classes of two-spin systems in the tree uniqueness regime: $\bullet$ all strictly anti-ferromagnetic two-spin systems (where both edge parameters $\beta,\gamma<1$), which cover the hardcore models and the anti-ferromagnetic Ising models; $\bullet$ general anti-ferromagnetic two-spin systems on regular graphs. Consequently, an optimal $O(n\log n)$ mixing time holds for these anti-ferromagnetic two-spin systems when the uniqueness condition is satisfied. These MLS and mixing time bounds hold for any bounded or unbounded maximum degree, and the constant factors in the bounds depend only on the gap to the uniqueness threshold. We prove this by showing a boosting theorem for MLS constant for distributions satisfying certain spectral independence and marginal stability properties.