Critical random graphs: Diameter and mixing time

Asaf Nachmias*, Yuval Peres

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

58 Scopus citations


Let C1 denote the largest connected component of the critical ErdosRényi random graph G(n, 1/n). We show that, typically, the diameter of C1 is of order n1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size n2/3 of p-bond percolation on any d-regular n-vertex graph where such clusters exist, provided that p(d - 1) ≤ 1 + O (n-1/3).

Original languageEnglish
Pages (from-to)1267-1286
Number of pages20
JournalAnnals of Probability
Issue number4
StatePublished - Jul 2008
Externally publishedYes


  • Mixing time
  • Percolation
  • Random graphs
  • Random walk


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