Addario-Berry, L., Broutin, N. and Goldschmidt, C. (Christina) (2012) The continuum limit of critical random graphs. Probability Theory and Related Fields, Vol.152 (No.3-4). pp. 367-406. doi:10.1007/s00440-010-0325-4

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Abstract

We consider the Erdos-Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn -4/3, for some fixed λ ε ℝ. We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n -1/3, converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n -1/3 converges in distribution to an absolutely continuous random variable with finite mean. © 2010 Springer-Verlag.

Item Type:	Journal Article
Subjects:	H Social Sciences > HA Statistics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Statistics
Journal or Publication Title:	Probability Theory and Related Fields
Publisher:	Springer
ISSN:	0178-8051
Official Date:	April 2012
Dates:	Date Event April 2012 Published
Volume:	Vol.152
Number:	No.3-4
Page Range:	pp. 367-406
DOI:	10.1007/s00440-010-0325-4
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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