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Probability Theory and Related Fields

, Volume 152, Issue 3–4, pp 367–406 | Cite as

The continuum limit of critical random graphs

  • L. Addario-BerryEmail author
  • N. Broutin
  • C. Goldschmidt
Article

Abstract

We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed \({\lambda \in \mathbb{R}}\) . 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.

Keywords

Random graphs Gromov–Hausdorff distance Scaling limits Continuum random tree Diameter 

Mathematics Subject Classification (2000)

05C80 60C05 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Projet algorithms, INRIA RocquencourtLe ChesnayFrance
  3. 3.Department of StatisticsUniversity of WarwickCoventryUK

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