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The continuum limit of critical random graphs
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  • Published: 12 October 2010

The continuum limit of critical random graphs

  • L. Addario-Berry1,
  • N. Broutin2 &
  • C. Goldschmidt3 

Probability Theory and Related Fields volume 152, pages 367–406 (2012)Cite this 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W., Montreal, QC, H3A 2K6, Canada

    L. Addario-Berry

  2. Projet algorithms, INRIA Rocquencourt, 78153, Le Chesnay, France

    N. Broutin

  3. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    C. Goldschmidt

Authors
  1. L. Addario-Berry
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  2. N. Broutin
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  3. C. Goldschmidt
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Corresponding author

Correspondence to L. Addario-Berry.

Additional information

L. Addario-Berry was supported by an NSERC Discovery Grant throughout the research and writing of this paper. C. Goldschmidt was funded by EPSRC Postdoctoral Fellowship EP/D065755/1.

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Addario-Berry, L., Broutin, N. & Goldschmidt, C. The continuum limit of critical random graphs. Probab. Theory Relat. Fields 152, 367–406 (2012). https://doi.org/10.1007/s00440-010-0325-4

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  • Received: 30 November 2009

  • Revised: 30 June 2010

  • Published: 12 October 2010

  • Issue Date: April 2012

  • DOI: https://doi.org/10.1007/s00440-010-0325-4

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Keywords

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

Mathematics Subject Classification (2000)

  • 05C80
  • 60C05
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