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May 2012 Mixing time of near-critical random graphs
Jian Ding, Eyal Lubetzky, Yuval Peres
Ann. Probab. 40(3): 979-1008 (May 2012). DOI: 10.1214/11-AOP647

Abstract

Let $\mathcal{C}_{1}$ be the largest component of the Erdős–Rényi random graph $\mathcal{G}(n,p)$. The mixing time of random walk on $\mathcal{C}_{1}$ in the strictly supercritical regime, p = c/n with fixed c > 1, was shown to have order log2n by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, p = (1 + ε)/n where λ = ε3n is bounded, Nachmias and Peres proved that the mixing time on $\mathcal{C}_{1}$ is of order n. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of $\mathcal{C}_{1}$ in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim.

In this paper, we show that for p = (1 + ε)/n with λ = ε3n → ∞ and λ = o(n), the mixing time on $\mathcal{C}_{1}$ is with high probability of order (n/λ)log2λ. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., p = (1 − ε)/n with λ as above].

Citation

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Jian Ding. Eyal Lubetzky. Yuval Peres. "Mixing time of near-critical random graphs." Ann. Probab. 40 (3) 979 - 1008, May 2012. https://doi.org/10.1214/11-AOP647

Information

Published: May 2012
First available in Project Euclid: 4 May 2012

zbMATH: 1243.05217
MathSciNet: MR2962084
Digital Object Identifier: 10.1214/11-AOP647

Subjects:
Primary: 05C80 , 05C81 , 60J10

Keywords: mixing time , Random graphs , Random walk

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 3 • May 2012
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