## The Annals of Probability

- Ann. Probab.
- Volume 40, Number 3 (2012), 979-1008.

### Mixing time of near-critical random graphs

Jian Ding, Eyal Lubetzky, and Yuval Peres

#### Abstract

Let be the largest component of the Erdős–Rényi random graph . The mixing time of random walk on in the strictly supercritical regime, *p* = *c*/*n* with fixed *c* > 1, was shown to have order log^{2}*n* by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, *p* = (1 + *ε*)/*n* where *λ* = *ε*^{3}*n* is bounded, Nachmias and Peres proved that the mixing time on is of order *n*. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim.

In this paper, we show that for *p* = (1 + *ε*)/*n* with *λ* = *ε*^{3}*n* → ∞ and *λ* = *o*(*n*), the mixing time on is with high probability of order (*n*/*λ*)log^{2}*λ*. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., *p* = (1 − *ε*)/*n* with *λ* as above].

#### Article information

**Source**

Ann. Probab., Volume 40, Number 3 (2012), 979-1008.

**Dates**

First available in Project Euclid: 4 May 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1336136056

**Digital Object Identifier**

doi:10.1214/11-AOP647

**Mathematical Reviews number (MathSciNet)**

MR2962084

**Zentralblatt MATH identifier**

1243.05217

**Subjects**

Primary: 05C80: Random graphs [See also 60B20] 05C81: Random walks on graphs 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Random graphs random walk mixing time

#### Citation

Ding, Jian; Lubetzky, Eyal; Peres, Yuval. Mixing time of near-critical random graphs. Ann. Probab. 40 (2012), no. 3, 979--1008. doi:10.1214/11-AOP647. https://projecteuclid.org/euclid.aop/1336136056