Cut-off for lamplighter chains on tori: dimension interpolation and Phase transition

Abstract

Given a finite, connected graph $\mathsf {G}$, the lamplighter chain on $\mathsf {G}$ is the lazy random walk $X^\diamond$ on the associated lamplighter graph $\mathsf {G}^\diamond =\mathbb {Z}_2 \wr \mathsf {G}$. The mixing time of the lamplighter chain on the torus $\mathbb {Z}_n^d$ is known to have a cutoff at a time asymptotic to the cover time of $\mathbb {Z}_n^d$ if $d=2$, and to half the cover time if $d \ge 3$. We show that the mixing time of the lamplighter chain on $\mathsf {G}_n(a)=\mathbb {Z}_n^2 \times \mathbb {Z}_{a \log n}$ has a cutoff at $\psi (a)$ times the cover time of $\mathsf {G}_n(a)$ as $n \rightarrow \infty$, where $\psi$ is an explicit weakly decreasing map from $(0,\infty )$ onto [1 / 2, 1). In particular, as $a > 0$ varies, the threshold continuously interpolates between the known thresholds for $\mathbb {Z}_n^2$ and $\mathbb {Z}_n^3$. Perhaps surprisingly, we find a phase transition (non-smoothness of $\psi$) at the point $a_*=\pi r_3 (1+\sqrt{2})$, where high dimensional behavior ($\psi (a)=1/2$ for all $a \ge a_*$) commences. Here $r_3$ is the effective resistance from 0 to $\infty$ in $\mathbb {Z}^3$.