# Cutoff at the entropic time for random walks on covered expander graphs

## TL;DR

This paper investigates the cutoff phenomenon for random walks on expander graphs, establishing conditions under which the walk exhibits cutoff at the entropic time, extending previous results from trees to more general graphs.

## Contribution

It generalizes the entropic cutoff result to a broader class of graphs by providing spectral conditions for cutoff, including anisotropic and lifted random walks.

## Key findings

- Cutoff occurs at the entropic time under certain spectral conditions.
- The results apply to anisotropic random walks on random regular graphs.
- The findings include non-reversible walks on lifted graphs.

## 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 $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparing the walk on $G_n$ to the walk on the infinite $d$-regular tree. If one can map another infinite transitive graph onto $G_n$, then we can improve the strategy by using a comparison with the random walk on this transitive graph (instead of that of the regular tree), and we obtain a lower bound of the form $1/h \log n$, where $h$ is the entropy rate associated with the walk on the transitive graph. We call this the entropic lower bound. It was recently proved that in the case of the tree, this entropic lower bound is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibit cutoff at the entropic time. In this paper, we provide a generalization of the result by providing a sufficient condition on the spectra the random walks on $G_n$ under which the random walk exhibit 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 non-reversible walks).