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A Point Process Describing the Component Sizes in the Critical Window of the Random Graph Evolution

Published online by Cambridge University Press:  01 July 2007

SVANTE JANSON  and
JOEL SPENCER
SVANTE JANSON
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden (e-mail: svante.janson@math.uu.se, http://www.math.uu.se/~svante/)
JOEL SPENCER
Affiliation:
Courant Institute, 251 Mercer St., New York, NY 10012, USA (e-mail: spencer@cs.nyu.edu, http://www.cs.nyu.edu/cs/faculty/spencer/)
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Abstract

We study a point process describing the asymptotic behaviour of sizes of the largest components of the random graph G(n, p) in the critical window, that is, for p = n−1 + λn−4/3, where λ is a fixed real number. In particular, we show that this point process has a surprising rigidity. Fluctuations in the large values will be balanced by opposite fluctuations in the small values such that the sum of the values larger than a small ϵ (a scaled version of the number of vertices in components of size greater than εn2/3) is almost constant.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 4 , July 2007 , pp. 631 - 658
Copyright
Copyright © Cambridge University Press 2007

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