Random walks on the random graph
Nathanael Berestycki, Eyal Lubetzky, Yuval Peres, Allan Sly

TL;DR
This paper demonstrates that on the giant component of an Erdős-Rényi random graph, the mixing time from a typical vertex is significantly faster and exhibits cutoff, with precise asymptotics related to the graph's local structure.
Contribution
It proves that starting from a uniform vertex accelerates mixing and causes cutoff in the giant component of Erdős-Rényi graphs, extending to graphs with prescribed degree sequences.
Findings
Typical mixing time is $O(rac{1}{ u f d}\, ext{log} )$
Cutoff phenomenon occurs for simple and non-backtracking random walks
Results extend to graphs with prescribed degree sequences
Abstract
We study random walks on the giant component of the Erd\H{o}s-R\'enyi random graph where for 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 . We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to and concentrates it (the cutoff phenomenon occurs): the typical mixing is at , where and are the speed of random walk and dimension of harmonic measure on a -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.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Markov Chains and Monte Carlo Methods · Mathematical Dynamics and Fractals
