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The Phase Transition in the Configuration Model

Published online by Cambridge University Press:  02 February 2012

OLIVER RIORDAN
OLIVER RIORDAN*
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK and Department of Mathematical Sciences, University of Memphis, TN 38152, USA (e-mail: riordan@maths.ox.ac.uk)
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Abstract

Let G = G(d) be a random graph with a given degree sequence d, such as a random r-regular graph where r ≥ 3 is fixed and n = |G| → ∞. We study the percolation phase transition on such graphs G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p. More generally, we study the emergence of a giant component in G(d) itself as d varies. We show that a single method can be used to prove very precise results below, inside and above the ‘scaling window’ of the phase transition, matching many of the known results for the much simpler model G(n, p). This method is a natural extension of that used by Bollobás and the author to study G(n, p), itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 265 - 299
Copyright
Copyright © Cambridge University Press 2012

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