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July 2008 Critical random graphs: Diameter and mixing time
Asaf Nachmias, Yuval Peres
Ann. Probab. 36(4): 1267-1286 (July 2008). DOI: 10.1214/07-AOP358

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 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).

Citation

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Asaf Nachmias. Yuval Peres. "Critical random graphs: Diameter and mixing time." Ann. Probab. 36 (4) 1267 - 1286, July 2008. https://doi.org/10.1214/07-AOP358

Information

Published: July 2008
First available in Project Euclid: 29 July 2008

zbMATH: 1160.05053
MathSciNet: MR2435849
Digital Object Identifier: 10.1214/07-AOP358

Subjects:
Primary: 05C80, 60C05, 82B43

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 4 • July 2008
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