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 of the lift is not reversible. The mixing time is w.h.p. , where h is a constant associated to , namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all n-lifts of . In the particular case where the base graph is a vertex with 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).
The author’s research is supported by a Ph.D. grant of ED386.
The author thanks Jonathan Hermon and an anonymous referee for many useful comments on a first preprint of this paper.
"Cutoff for random lifts of weighted graphs." Ann. Probab. 50 (1) 304 - 338, January 2022. https://doi.org/10.1214/21-AOP1534