Mixing times of spin systems on dynamical percolation

We study the mixing times of stochastic spin systems corresponding to nearest-neighbour Glauber dynamics on dynamical percolation, defined on $d$-dimensional torus of side-length $N$. In this model, the status of each edge (open or closed) updates independently at rate $\lambda>0$, according to $\mathrm{Ber}(p)$ samples. Simultaneously, the spin of each site updates at rate $1$ according to Glauber dynamics on the environment restricted to open edges. We show that for a relatively general class of nearest-neighbour systems, as long as $p<p_c(d)$, for any temperature, if $\lambda$ is sufficiently small, the mixing time is of order $\frac{\log N}{\lambda}$. This Markov chain is non-reversible, and the proof is obtained by developing a particular coupling that couples together local configurations whenever the environment behaves well.