The large deviation principle for the Erdo{double acute}s-Rényi random graph

Sourav Chatterjee, Srinivasa Varadhan

Research output: Contribution to journalArticle

Abstract

What does an Erdo{double acute}s-Rényi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovász and coauthors and Szemerédi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.

Original languageEnglish (US)
Pages (from-to)1000-1017
Number of pages18
JournalEuropean Journal of Combinatorics
Volume32
Issue number7
DOIs
StatePublished - Oct 2011

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Large Deviation Principle
Graph theory
Random Graphs
Acute
Phase transitions
Topology
Regularity Lemma
Rare Events
Graph in graph theory
Large Deviations
Triangle
Phase Transition
Infinity
Tend
Formulation

ASJC Scopus subject areas

  • Geometry and Topology
  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

The large deviation principle for the Erdo{double acute}s-Rényi random graph. / Chatterjee, Sourav; Varadhan, Srinivasa.

In: European Journal of Combinatorics, Vol. 32, No. 7, 10.2011, p. 1000-1017.

Research output: Contribution to journalArticle

Chatterjee, Sourav ; Varadhan, Srinivasa. / The large deviation principle for the Erdo{double acute}s-Rényi random graph. In: European Journal of Combinatorics. 2011 ; Vol. 32, No. 7. pp. 1000-1017.
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