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Diameters in Supercritical Random Graphs Via First Passage Percolation

Published online by Cambridge University Press:  05 October 2010

JIAN DING ,
JEONG HAN KIM ,
EYAL LUBETZKY  and
YUVAL PERES
JIAN DING
Affiliation:
Department of Statistics, UC Berkeley, Berkeley, CA 94720, USA (e-mail: jding@stat.berkeley.edu)
JEONG HAN KIM
Affiliation:
Department of Mathematics, Yonsei University, Seoul 120-749, Korea and National Institute for Mathematical Sciences, Daejeon 305-340, Korea (e-mail: jehkim@yonsei.ac.kr)
EYAL LUBETZKY
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA (e-mail: eyal@microsoft.com, peres@microsoft.com)
YUVAL PERES
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA (e-mail: eyal@microsoft.com, peres@microsoft.com)
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Abstract

We study the diameter of 1, the largest component of the Erdős–Rényi random graph (n, p) in the emerging supercritical phase, i.e., for p = where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of .

We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of 1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of 1 is w.h.p. asymptotic to , and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to .

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