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 $$\mathbb {Z}^3$$ .