UNIVERSALITY OF CUTOFF FOR THE ISING MODEL

• 成果类型: Article
• 署名作者: Lubetzky, Eyal; Sly, Allan
• 署名单位: New York University; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1146
• 发表日期: 2017
• 页码: 3664-3696
• 关键词: logarithmic sobolev inequalities; finite markov-chains; one-phase region; glauber dynamics; spin systems; random-walks; lattice; equilibrium; set

摘要

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 1970s. 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 -> 0 as beta -> 0), whereas the uniform starting state is at least 2 - epsilon beta times faster.