CUTOFF FOR THE SWENDSEN-WANG DYNAMICS ON THE LATTICE

• 成果类型: Article
• 署名作者: Nam, Danny; Sly, Allan
• 署名单位: Princeton University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1344
• 发表日期: 2019
• 页码: 3705-3761
• 关键词: random-cluster; ising-model; mixing properties; algorithm

摘要

We study the Swendsen-Wang dynamics for the q-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen-Wang dynamics is a nonlocal Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work, we establish cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at sufficiently high temperatures, proving that it exhibits a sharp transition from unmixed to well mixed. In particular, we show that at high enough temperatures the Swendsen-Wang dynamics on the torus (Z/n/Z)(d) has cutoff at time d/2 (- log(1 - gamma))(-1) logn, where gamma(beta) is the spectral gap of the infinite-volume dynamics.