The continuum limit of critical random graphs

• Published in: Probability theory and related fields Vol. 152; no. 3-4; pp. 367 - 406
• Main Authors: Addario-Berry, L., Broutin, N., Goldschmidt, C.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.04.2012; Springer Nature B.V; Springer Verlag
• Subjects: Brownian motion; Combinatorics; Continuity (mathematics); Continuums; Convergence; Economics; Finance; Graph theory; Graphs; Insurance; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Metric space; Operations Research/Decision Theory; Phase transitions; Probability; Probability theory; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Random walk; Statistics for Business; Studies; Theoretical; Trees; 60C05; Random graphs; Continuum random tree; Gromov–Hausdorff distance; 05C80; Scaling limits; Diameter
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-010-0325-4

We consider the Erdős–Rényi random graph G ( n , p ) inside the critical window, that is when p  = 1/ n  + λ n −4/3 , for some fixed . 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.