Mixing time of random walk on dynamical random cluster
成果类型：
Article
署名作者：
Lelli, Andrea; Stauffer, Alexandre
署名单位：
University of Bath; University of London; King's College London
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-024-01262-8
发表日期：
2024
页码：
981-1043
关键词：
摘要：
We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n2/mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n<^>2/\mu $$\end{document}. In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.