The large deviation principle for the Erdős-Rényi random graph

Abstract

What does an Erdős-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.