Critical Random Graphs: Diameter and Mixing Time 1

Let C1 denote the largest connected component of the critical Erdős– Rényi 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).

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