Asymptotic normality of the size of the giant component via a random walk

Details Asymptotic normality of the size of the giant component via a random walk Asymptotic normality of the size of the giant component via a random walk

2010-10-21

arxiv.org/abs/1010.4595v2

J. Combinatorial Theory B 102 (2012), 53--61

Mathematics

Probability

11 pages; slightly expanded, reference added

Scientific paper

10.1016/j.jctb.2011.04.003

In this paper we give a simple new proof of a result of Pittel and Wormald concerning the asymptotic value and (suitably rescaled) limiting distribution of the number of vertices in the giant component of $G(n,p)$ above the scaling window of the phase transition. Nachmias and Peres used martingale arguments to study Karp's exploration process, obtaining a simple proof of a weak form of this result. We use slightly different martingale arguments to obtain a much sharper result with little extra work.

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