The Annals of Probability

Critical random graphs: Diameter and mixing time

Asaf Nachmias and Yuval Peres

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Let $\mathcal{C}_{1}$ denote the largest connected component of the critical Erdős–Rényi random graph $G(n,{\frac{1}{n}})$. We show that, typically, the diameter of $\mathcal{C}_{1}$ is of order n1/3 and the mixing time of the lazy simple random walk on $\mathcal{C}_{1}$ 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).

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Ann. Probab., Volume 36, Number 4 (2008), 1267-1286.

First available in Project Euclid: 29 July 2008

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 82B43: Percolation [See also 60K35] 60C05: Combinatorial probability

Percolation random graphs random walk mixing time


Nachmias, Asaf; Peres, Yuval. Critical random graphs: Diameter and mixing time. Ann. Probab. 36 (2008), no. 4, 1267--1286. doi:10.1214/07-AOP358.

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