Critical random graphs: Diameter and mixing time

Abstract

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 $n^{1/3}$ 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 $n^{2/3}$ of $p$-bond percolation on any $d$-regular $n$-vertex graph where such clusters exist, provided that $p(d-1)\le1+O(n^{-1/3})$.

Published in at http://dx.doi.org/10.1214/07-AOP358 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)