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It is a fact simple to establish that the mixing time of the simple randomwalk on a d-regular graph $G_n$ with n vertices is asymptotically bounded frombelow by $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparingthe walk on $G_n$ to the walk on the infinite $d$-regular tree. If one can mapanother infinite transitive graph onto $G_n$, then we can improve the strategyby using a comparison with the random walk on this transitive graph (instead ofthat of the regular tree), and we obtain a lower bound of the form $1/h \logn$, where $h$ is the entropy rate associated with the walk on the transitivegraph. We call this the entropic lower bound. It was recently proved that inthe case of the tree, this entropic lower bound is sharp when graphs haveminimal spectral radius and thus that in that case the random walk exhibitcutoff at the entropic time. In this paper, we provide a generalization of theresult 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. Itapplies notably to anisotropic random walks on random $d$-regular graphs and torandom walks on random $n$-lifts of a base graph (including non-reversiblewalks).

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