It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph Gn with n vertices is asymptotically bounded from below by d/((d−2)log(d−1))logn
Nearby in the stack
. Such a bound is obtained by comparing the walk on
Gn
to the walk on the infinite
d
-regular tree. If one can map another infinite transitive graph onto
Gn
, 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/hlogn
, 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
Gn
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).