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CUTOFF AT THE ENTROPIC TIME FOR RANDOM WALKS ON COVERED EXPANDER GRAPHS
Published online by Cambridge University Press: 09 February 2021
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).
MSC classification
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1571 - 1616
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
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