Mixing time of near-critical random graphs

Jian Ding, Eyal Lubetzky, Yuval Peres

Research output: Contribution to journalArticlepeer-review


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].

Original languageEnglish (US)
Pages (from-to)979-1008
Number of pages30
JournalAnnals of Probability
Issue number3
StatePublished - May 2012


  • Mixing time
  • Random graphs
  • Random walk

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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