Logarithmic Mixing of Random Walks on Dynamical Random Cluster Models

2026-05-07

AI Summary

This study investigates the mixing time of random walks on stochastic dynamically evolving graphs, where the underlying graph is generated by a dynamic random cluster model whose evolution depends on global connectivity, thereby inducing strong coupling between the walk and its environment. Focusing on the subcritical regime with a random $d$-regular base graph, the authors combine coupling arguments with path-counting techniques to effectively control the globally dependent evolution of the environment along typical trajectories. They establish, for the first time in a strongly environment-dependent dynamic graph model, that when the edge-update rate $\mu \geq \varepsilon \log n$, the joint process mixes in $\Theta(\log n)$ time—matching the mixing time of simple random walk on static random regular graphs. This demonstrates that a rapidly evolving environment does not impede mixing, thereby transcending the limitations of traditional static-graph assumptions.

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 $μ>0$ 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.

Problem

• random walks
• dynamical random cluster
• mixing time
• Glauber dynamics
• subcritical regime

Innovation

• dynamical random-cluster model
• random walk
• mixing time
• coupling argument
• subcritical regime