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CUTOFF AT THE ENTROPIC TIME FOR RANDOM WALKS ON COVERED EXPANDER GRAPHS

Part of: Markov processes Graph theory

Published online by Cambridge University Press:  09 February 2021

Charles Bordenave  and
Hubert Lacoin Open the ORCID record for Hubert Lacoin [Opens in a new window]
Charles Bordenave
Affiliation:
Institut de Mathématiques de Marseille Université d'Aix-Marseille 163 avenue de Luminy - Case 907 13288 Marseille Cedex 9 - France (charles.bordenave@univ-amu.fr)
Hubert Lacoin
Affiliation:
IMPA Estrada Dona Castorina 110 Rio de Janeiro / Brasil22460-320 tel: +55 21 2529 5118 (lacoin@impa.br)
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Abstract

It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ . Such a bound is obtained by comparing the walk on $G_n$ to the walk on d-regular tree $\mathcal{T}_d$ . If one can map another transitive graph $\mathcal{G} $ onto $G_n$ , then we can improve the strategy by using a comparison with the random walk on $\mathcal{G} $ (instead of that of $\mathcal{T} _d$ ), and we obtain a lower bound of the form $\frac {1}{\mathfrak{h} }\log n$ , where $\mathfrak{h} $ is the entropy rate associated with $\mathcal{G} $ . We call this the entropic lower bound.

It was recently proved that in the case $\mathcal{G} =\mathcal{T} _d$ , this entropic lower bound (in that case $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ ) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on $G_n$ under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random d-regular graphs and to random walks on random n-lifts of a base graph (including nonreversible walks).

Keywords

expander graphsmixing timespectral gapRamanujan graphentropic time

MSC classification

Primary: 05C81: Random walks on graphs 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1571 - 1616
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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