Universality of cutoff for the Ising model

TL;DR

This paper proves that the Ising model exhibits cutoff at high enough temperatures across various geometries, with precise cutoff time and window, using a new information percolation framework.

Contribution

It introduces a new framework of information percolation to establish cutoff for the Ising model on any graph with bounded degree at sufficiently high temperatures, extending beyond lattices.

Findings

Cutoff occurs when $eta<rac{ ext{constant}}{d}$ for graphs with maximum degree $d$.

Cutoff location is when the sum of squared magnetizations drops to 1.

Cutoff window is of order 1, similar to the zero-temperature case.

Abstract

On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature $\beta$ is small enough, via classical results of Dobrushin and of Holley in the 1970's. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed $\beta>0$, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree $d$, the Ising model has cutoff provided that $\beta<\kappa/d$ for some absolute constant $\kappa$ (a result which, up to the value of $\kappa$, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is $O(1)$, just as when $\beta=0$. Finally, the mixing time from almost every initial state is not more than a factor of $1+\epsilon_\beta$ faster then the worst one (with $\epsilon_\beta\to0$ as $\beta\to 0$), whereas the uniform starting state is at least $2-\epsilon_\beta$ times faster.