Cutoff for General Spin Systems with Arbitrary Boundary Conditions

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/ n ℤ) d for any d ≥ ;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 the stochastic Ising model on any sequence of bounded‐degree graphs with subexponential 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 nonmonotone spin systems at high temperatures, including the gas hard‐core model, the Potts model, the antiferromagnetic Potts model, and the coloring model. © 2014 Wiley Periodicals, Inc.

Locations

• Communications on Pure and Applied Mathematics - View
• arXiv (Cornell University) - View - PDF