Glauber dynamics on nonamenable graphs: Boundary conditions and mixing time

TL;DR

This paper investigates how boundary conditions affect the mixing time of the stochastic Ising model on nonamenable graphs, showing that (+)-boundary conditions can significantly accelerate convergence at low temperatures.

Contribution

It demonstrates that for certain nonamenable graphs, (+)-boundary conditions ensure a uniformly positive spectral gap and linear mixing time, extending previous analyses to a broader class of graphs.

Findings

Positive spectral gap under (+)-boundary conditions at low temperatures

Mixing time is at most linear in the number of vertices n

Existence of a graph with exponential mixing time under free boundary conditions

Abstract

We study the stochastic Ising model on finite graphs with n vertices and bounded degree and analyze the effect of boundary conditions on the mixing time. We show that for all low enough temperatures, the spectral gap of the dynamics with (+)-boundary condition on a class of nonamenable graphs, is strictly positive uniformly in n. This implies that the mixing time grows at most linearly in n. The class of graphs we consider includes hyperbolic graphs with sufficiently high degree, where the best upper bound on the mixing time of the free boundary dynamics is polynomial in n, with exponent growing with the inverse temperature. In addition, we construct a graph in this class, for which the mixing time in the free boundary case is exponentially large in n. This provides a first example where the mixing time jumps from exponential to linear in n while passing from free to (+)-boundary condition. These results extend the analysis of Martinelli, Sinclair and Weitz to a wider class of nonamenable graphs.