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.

翻译：我们研究了动态演化图上的随机游走，其中环境由基础图边集的一个随时间变化的子集给出。具体来说，遵循Lelli和Stauffer最近引入的框架，我们考虑一个与动态随机团簇环境相互作用的随机游走。在该环境中，边以速率$μ>0$根据参数为$p$和$q$的Glauber动力学更新，而游走者以速率1移动，但只能穿越在移动时刻存在的边。这种设定引入了游走与环境之间的强依赖性，因为边更新概率依赖于全局连通结构。我们关注基础图为随机$d$正则图且参数处于亚临界区域$p < p_{\mathrm{u}}(q, d)$（已知该区域中Glauber动力学快速混合）的情况。我们的主要结果是证明：对于任意$\varepsilon >0$和所有$q \ge 1$，当$μ\geq \varepsilon \log n$时，联合过程的混合时间（在连续时间中）为$Θ(\log n)$。这与静态随机正则图上简单随机游走的混合时间相匹配，表明在该区域中演化环境不会减缓混合速度。我们的证明基于耦合论证，通过沿典型轨迹控制环境结构，利用路径计数技术克服边动力学中的依赖性。