Mixing time of random walk on dynamical random cluster

Abstract

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 μ\\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/μ\\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.