  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 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 n^2/\mu. In our proof we construct of a
non-Markovian coupling through a multi-scale analysis of the environment, which
we believe could be more widely applicable.
