Random walks on the random graph

Abstract

We study random walks on the giant component of the Erdős-Rényi random graph G(n, p) where p = λ/n for λ > 1 fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order log 2 n. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to O(log n) and concentrates it (the cutoff phenomenon occurs): the typical mixing is at (νd) −1 log n ± (log n) 1/2+o(1) , where ν and d are the speed of random walk and dimension of harmonic measure on a Poisson(λ)-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.

Invariant measures, Hausdorff dimension and dimension drop of some harmonic measures on Galton-Watson trees

“…For a more concrete interpretation of this phenomenon, see [13,Corollary 5.3] and [12,Theorem 9.9] which show that with high probability, the random walk is confined to an exponentially smaller subtree. Another application of this result is given in [3], where the authors prove a cut-off phenomenon for the mixing time of the random walk on the giant component of an Erdős-Rényi random graph, started from a typical point.…”

Comparing mixing times on sparse random graphs

“…the NBRW also exhibits cutoff at h −1 Y log n under similar moment assumptions (in [4] the Gaussian tail of the distance profile within the cutoff window was further established). Here we extend the arguments of [7] to provide the analogous cutoff result for the SRW from a worst-case starting point, as well as compare these cutoff locations.…”

“…It was shown in [7] that, when the initial vertex v 1 is fixed (independently of the graph) and the degree distribution (p k ) k≥1 satisfies suitable moment assumptions, with high probability (w.h.p.) the SRW has cutoff at time h −1 X log n whereas the NBRW has cutoff at time h −1 Y log n. Comparing these two mixing times was left open.…”

“…From a worst-case initial vertex, it was showed independently in [4,7] that w.h.p. the NBRW also exhibits cutoff at h −1 Y log n under similar moment assumptions (in [4] the Gaussian tail of the distance profile within the cutoff window was further established).…”

“…We first recall some key facts established in [7] in the analysis of SRW on sparse irregular random graphs. For a given graph G = (V, E) on n vertices and vertex x ∈ V , an integer t > 0 and constants γ, ε > 0, define the event…”