Logarithmic Mixing of Random Walks on Dynamical Random Cluster Models

TL;DR

This paper proves that random walks on dynamically evolving random regular graphs mix in logarithmic time when edges update rapidly, matching static graph mixing times despite environment dependencies.

Contribution

It establishes that fast edge updates in a dynamical random-cluster environment lead to mixing times comparable to static graphs, using novel coupling and path-count techniques.

Findings

Mixing time is Θ(log n) for μ ≥ ε log n in the subcritical regime.

Environment dependencies are effectively controlled by the coupling argument.

Fast edge updates do not slow down the mixing compared to static graphs.

Abstract

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 $\mu>0$ 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 $\Theta(\log n)$ (in continuous time) whenever $\mu\geq \varepsilon \log n$. 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.