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The continuum limit of critical random graphs
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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 ISSN 0178-8051.
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Official URL: http://dx.doi.org/10.1007/s00440-010-0325-4
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 | ||||
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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: |
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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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