Mixing time of random walk on dynamical random cluster

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 μ 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 n 2 / μ . 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.