Electronic Communications in Probability

Cutoff for Ramanujan graphs via degree inflation

Jonathan Hermon

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Recently Lubetzky and Peres showed that simple random walks on a sequence of $d$-regular Ramanujan graphs $G_n=(V_n,E_n)$ of increasing sizes exhibit cutoff in total variation around the diameter lower bound $\frac{d} {d-2}\log _{d-1}|V_n| $. We provide a different argument under the assumption that for some $r(n) \gg 1$ the maximal number of simple cycles in a ball of radius $r(n)$ in $G_n$ is uniformly bounded in $n$.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 45, 10 pp.

Received: 26 February 2017
Accepted: 5 July 2017
First available in Project Euclid: 15 August 2017

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Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C12: Distance in graphs 05C81: Random walks on graphs

cutoff Ramanujan graphs degree inflation

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Hermon, Jonathan. Cutoff for Ramanujan graphs via degree inflation. Electron. Commun. Probab. 22 (2017), paper no. 45, 10 pp. doi:10.1214/17-ECP72. https://projecteuclid.org/euclid.ecp/1502762751

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