January 2022 Cutoff for random lifts of weighted graphs
Guillaume Conchon-Kerjan
Author Affiliations +
Ann. Probab. 50(1): 304-338 (January 2022). DOI: 10.1214/21-AOP1534


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 G of the lift is not reversible. The mixing time is w.h.p. tmix=h1logn, where h is a constant associated to G, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all n-lifts of G. In the particular case where the base graph is a vertex with 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).

Funding Statement

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.


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Guillaume Conchon-Kerjan. "Cutoff for random lifts of weighted graphs." Ann. Probab. 50 (1) 304 - 338, January 2022. https://doi.org/10.1214/21-AOP1534


Received: 1 August 2019; Revised: 1 July 2020; Published: January 2022
First available in Project Euclid: 23 February 2022

MathSciNet: MR4385128
zbMATH: 1486.05284
Digital Object Identifier: 10.1214/21-AOP1534

Primary: 05C81 , 60B10
Secondary: 05C05 , 60J10

Keywords: Cutoff , periodic trees , Random walks

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 1 • January 2022
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