## The Annals of Probability

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

### Random walks on the random graph

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

#### Abstract

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

**Source**

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

**Dates**

Received: May 2015

Revised: October 2016

First available in Project Euclid: 5 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1517821227

**Digital Object Identifier**

doi:10.1214/17-AOP1189

**Mathematical Reviews number (MathSciNet)**

MR3758735

**Zentralblatt MATH identifier**

06865127

**Subjects**

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]

**Keywords**

Random walks random graphs cutoff phenomenon Markov chain mixing times

#### Citation

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. https://projecteuclid.org/euclid.aop/1517821227