Logarithmic Mixing of Random Walks on Dynamical Random Cluster Models
This work addresses the fundamental question of how dynamic environments affect random walk mixing, providing a tight result for a natural dependent model.
The paper studies mixing times of random walks on dynamic random cluster models on random d-regular graphs. It shows that in the subcritical regime, the mixing time is Θ(log n) when the environment update rate is at least ε log n, matching the static case.
We study random walks on dynamically evolving graphs, where the environment is given by a time-dependent subset of the edges of an underlying graph. Concretely, following the recently introduced framework of Lelli and Stauffer, we consider a random walk interacting with a dynamical random-cluster environment, in which edges are updated with rate $μ>0$ according to Glauber dynamics with parameters $p$ and $q$, and the walker moves at rate 1 but may only traverse edges that are present at the time of the move. This setting introduces strong dependencies between the walk and the environment, as edge-update probabilities depend on the global connectivity structure. We focus on the case where the underlying graph is a random $d$-regular graph and the parameters lie in the subcritical regime $p < p_{\mathrm{u}}(q, d)$ where it is known that the Glauber dynamics mixes quickly. Our main result is to show that for any $\varepsilon >0$ and all $q \ge 1$, for all $p$ in the subcritical regime, the mixing time of the joint process is $Θ(\log n)$ (in continuous time) whenever $μ\geq \varepsilon \log n$. This matches the mixing time of the simple random walk on a static random regular graph, showing that in this regime the evolving environment does not slow down mixing. Our proof is based on a coupling argument that uses path-count techniques to overcome the dependencies in the edge dynamics by controlling the structure of the environment along typical trajectories.