Cutoff for random lifts of weighted graphs

Abstract

We prove the cutoff phenomenon for the random walk on random n-lifts of finite weighted graphs, even when the random walk on the base graph $\mathcal{G}$ of the lift is not reversible. The mixing time is w.h.p. ${\mathit{t}}_{\mathrm{mix}}={\mathit{h}}^{-1}log\mathit{n}$, where h is a constant associated to $\mathcal{G}$, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all n-lifts of $\mathcal{G}$. In the particular case where the base graph is a vertex with $\mathit{d}/2$ loops, d even, we obtain a cutoff for a d-regular random graph, as did Lubetzky and Sly in (Duke Math. J. 153 (2010) 475–510) (with a slightly different distribution on d-regular graphs, but the mixing time is the same).