The Annals of Probability

Random walks on the random graph

Nathanaël Berestycki, Eyal Lubetzky, Yuval Peres, and Allan Sly

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We study random walks on the giant component of the Erdős–Rényi random graph $\mathcal{G}(n,p)$ where $p=\lambda/n$ for $\lambda>1$ 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 $(\nu\mathbf{d})^{-1}\log n\pm(\log n)^{1/2+o(1)}$, where $\nu$ and $\mathbf{d}$ are the speed of random walk and dimension of harmonic measure on a $\operatorname{Poisson}(\lambda)$-Galton–Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the nonbacktracking random walk.

Article information

Ann. Probab., Volume 46, Number 1 (2018), 456-490.

Received: May 2015
Revised: October 2016
First available in Project Euclid: 5 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G50: Sums of independent random variables; random walks 05C80: Random graphs [See also 60B20]

Random walks random graphs cutoff phenomenon Markov chain mixing times


Berestycki, Nathanaël; Lubetzky, Eyal; Peres, Yuval; Sly, Allan. Random walks on the random graph. Ann. Probab. 46 (2018), no. 1, 456--490. doi:10.1214/17-AOP1189.

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