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)≤1+O(n^−1/3).