January 2022 Universality of cutoff for graphs with an added random matching
Jonathan Hermon, Allan Sly, Perla Sousi
Author Affiliations +
Ann. Probab. 50(1): 203-240 (January 2022). DOI: 10.1214/21-AOP1532


We establish universality of cutoff for simple random walk on a class of random graphs defined as follows. Given a finite graph G=(V,E) with |V| even we define a random graph G=(V,EE) obtained by picking E to be the (unordered) pairs of a random perfect matching of V. We show that for a sequence of such graphs Gn of diverging sizes and of uniformly bounded degree, if the minimal size of a connected component of Gn is at least 3 for all n, then the random walk on Gn exhibits cutoff w.h.p. This provides a simple generic operation of adding some randomness to a given graph, which results in cutoff.

Funding Statement

Jonathan Hermon’s research was supported by an NSERC grant. Allan Sly’s research was partially supported by NSF Grant DMS-1855527, a Simons Investigator grant and a MacArthur Fellowship. Perla Sousi’s research was supported by the Engineering and Physical Sciences Research Council: EP/R022615/1.


The authors would like to thank Persi Diaconis, Balázs Gerencsér, David Levin and Evita Nestoridi for useful discussions.


Download Citation

Jonathan Hermon. Allan Sly. Perla Sousi. "Universality of cutoff for graphs with an added random matching." Ann. Probab. 50 (1) 203 - 240, January 2022. https://doi.org/10.1214/21-AOP1532


Received: 1 September 2020; Revised: 1 April 2021; Published: January 2022
First available in Project Euclid: 23 February 2022

MathSciNet: MR4385126
zbMATH: 1486.05278
Digital Object Identifier: 10.1214/21-AOP1532

Primary: 60F05 , 60G50 , 60J10
Secondary: 05C80 , 05C81

Keywords: Cutoff , Entropy , mixing time , quasi trees , random graph

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 1 • January 2022
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