Cutoff for general spin systems with arbitrary boundary conditions

TL;DR

This paper extends the understanding of the cutoff phenomenon in Markov chains, particularly for spin systems like the Ising model, to more general geometries and boundary conditions, providing new criteria and explicit cutoff locations.

Contribution

It generalizes previous results by establishing cutoff criteria for spin systems on arbitrary graphs and boundary conditions, including non-monotone models at high temperatures.

Findings

Cutoff occurs for spin systems on graphs with sub-exponential growth under bounded inverse log-Sobolev constants.

Explicit cutoff location identified for lattices with homogeneous boundary conditions.

Cutoff results extended to non-monotone high-temperature spin systems like Potts and coloring models.

Abstract

The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the $d$-dimensional torus $(Z/nZ)^d$ for any $d\geq 1$. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log-Sobolev constant of the Glauber dynamics. In particular, we show there is cutoff for stochastic Ising on any sequence of bounded-degree graphs with sub-exponential growth under arbitrary external fields provided the inverse log-Sobolev constant is bounded. For lattices with homogenous boundary, such as all-plus, we identify the cutoff location explicitly in terms of spectral gaps of infinite-volume dynamics on half-plane intersections. Analogous results establishing cutoff are obtained for non-monotone spin-systems at high temperatures, including the gas hard-core model, the Potts model, the anti-ferromagnetic Potts model and the coloring model.