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January 2018 Random walks on the random graph
Nathanaël Berestycki, Eyal Lubetzky, Yuval Peres, Allan Sly
Ann. Probab. 46(1): 456-490 (January 2018). DOI: 10.1214/17-AOP1189


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.


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Nathanaël Berestycki. Eyal Lubetzky. Yuval Peres. Allan Sly. "Random walks on the random graph." Ann. Probab. 46 (1) 456 - 490, January 2018.


Received: 1 May 2015; Revised: 1 October 2016; Published: January 2018
First available in Project Euclid: 5 February 2018

zbMATH: 06865127
MathSciNet: MR3758735
Digital Object Identifier: 10.1214/17-AOP1189

Primary: 05C80 , 60B10 , 60G50 , 60J10

Keywords: Cutoff phenomenon , Markov chain mixing times , Random graphs , Random walks

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 1 • January 2018
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