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

Abstract

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 n log(d−1) . Such a bound is obtained by comparing the walk on Gn to the walk on d-regular tree T d . If one can map another transitive graph G onto Gn, then we can improve the strategy by using a comparison with the random walk on G (instead of that of T d ), and we obtain a lower bound of the form 1 h log n, where h is the entropy rate associated with G. We call this the entropic lower bound. It was recently proved that in the case G = T d , this entropic lower bound (in that case d d−2 log n log(d−1) )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.

Cutoff for Almost All Random Walks on Abelian Groups

“…In this set-up, a family of transition matrices chosen from a certain family of distributions is shown to, whp, give rise to a sequence of Markov chains which exhibits cutoff. A few notable examples include random birth and death chains [17,45], the simple or non-backtracking random walk on various models of sparse random graphs, including random regular graphs [37], random graphs with given degrees [5,6,7,8], the giant component of the Erdős-Rényi random graph [7] (where the authors consider mixing from a 'typical' starting point) and a large family of sparse Markov chains [8], as well as random walks on a certain generalisation of Ramanujan graphs [9] and random lifts [9,12].…”

“…The cutoff time is then shown to be (up to smaller order terms) the time at which the entropy of the auxiliary process equals the entropy of the invariant distribution of the original Markov chain. It is a relatively new technique, and has been used recently in [7,8,9,12]. For 'most' regimes of k, this is the case for us too; further, for the non-Abelian groups considered in [24] we use a similar idea.…”

“…We use an 'entropic method', as mentioned in §1.3; cf [7,8,9,12]. The method is fairly general; we now explain the specific application in a little more depth.…”

Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues

“…[23,22,15,7]). Theorem 1.4 was recently proved independently in [4]. The relation between the optimality of the almost-diameter and the Ramanujan property is also studied in different contexts.We already mentioned the work of Sardari ( [27]) and Lubetzky and Peres ( [23]), but it is also closely related to the work of Parzanchevski and Sarnak about Golden Gates ( [26]) and the general results of Ghosh, Gorodnik and Nevo ( [13]).…”

On Sarnak's Density Conjecture and its Applications

“…Recently, Sarnak made a density conjecture which is an approximation to the very naive Ramanujan property and should serve as a replacement of it for applications. Some instances of this general idea were previously given for hyperbolic surfaces ( [52,27]) and for graphs ( [6,35]). Our goal here is to give a general framework for the proof of similar density conjectures and their use in applications.…”