Mixing and relaxation time for random walk on wreath product graphs

Abstract

Suppose that $G$ and $H$ are finite, connected graphs, $G$ regular, $X$ is a lazy random walk on $G$ and $Z$ is a reversible ergodic Markov chain on $H$. The generalized lamplighter chain $X^{\diamond}$ associated with $X$ and $Z$ is the random walk on the wreath product $H \wr G$, the graph whose vertices consist of pairs $(f,x)$ where $f=(f_v)_{v\in V(G)}$ is a labeling of the vertices of $G$ by elements of $H$ and $x$ is a vertex in $G$. In each step, $^{\diamond}$* moves from a configuration $(f,x)$ by updating x to y using the transition rule of $X$ and then independently updating both $f_x$ and $f_y$ according to the transition probabilities on $H$; $f_z$ for $z$ different of $x,y$ remains unchanged. We estimate the mixing time of $X^{\diamond}$ in terms of the parameters of $H$ and $G$. Further, we show that the relaxation time of $X^{\diamond}$ is the same order as the maximal expected hitting time of $G$ plus $|G|$ times the relaxation time of the chain on $H$.