### Abstract

Let C _{1} be the largest component of the Erdo{double acute}s-Rényi random graph G(n,p). The mixing time of random walk on C _{1} 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 andWormald. In the critical window, p = (1+ε)/n where λ = ε ^{3}n is bounded, Nachmias and Peres proved that the mixing time on C _{1} 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 C1 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 C _{1} 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].

Original language | English (US) |
---|---|

Pages (from-to) | 979-1008 |

Number of pages | 30 |

Journal | Annals of Probability |

Volume | 40 |

Issue number | 3 |

DOIs | |

State | Published - May 2012 |

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### Keywords

- Mixing time
- Random graphs
- Random walk

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*40*(3), 979-1008. https://doi.org/10.1214/11-AOP647

**Mixing time of near-critical random graphs.** / Ding, Jian; Lubetzky, Eyal; Peres, Yuval.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 40, no. 3, pp. 979-1008. https://doi.org/10.1214/11-AOP647

}

TY - JOUR

T1 - Mixing time of near-critical random graphs

AU - Ding, Jian

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2012/5

Y1 - 2012/5

N2 - Let C 1 be the largest component of the Erdo{double acute}s-Rényi random graph G(n,p). The mixing time of random walk on C 1 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 andWormald. In the critical window, p = (1+ε)/n where λ = ε 3n is bounded, Nachmias and Peres proved that the mixing time on C 1 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 C1 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 3n → ∞ and λ = o(n), the mixing time on C 1 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].

AB - Let C 1 be the largest component of the Erdo{double acute}s-Rényi random graph G(n,p). The mixing time of random walk on C 1 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 andWormald. In the critical window, p = (1+ε)/n where λ = ε 3n is bounded, Nachmias and Peres proved that the mixing time on C 1 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 C1 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 3n → ∞ and λ = o(n), the mixing time on C 1 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].

KW - Mixing time

KW - Random graphs

KW - Random walk

UR - http://www.scopus.com/inward/record.url?scp=84863000953&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863000953&partnerID=8YFLogxK

U2 - 10.1214/11-AOP647

DO - 10.1214/11-AOP647

M3 - Article

AN - SCOPUS:84863000953

VL - 40

SP - 979

EP - 1008

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 3

ER -