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

  • Authors:

  • Béla Bollobás;Oliver Riordan

  • Affiliations:

  • Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA;Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

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Abstract

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.