The large deviation principle for the Erd\H{o}s-R\'enyi random graph

TL;DR

This paper establishes a large deviation principle for Erdős-Rényi graphs as the number of vertices grows, using graph limit theory and Szemerédi's regularity lemma, revealing phase transitions in triangle counts.

Contribution

It introduces a large deviation framework for Erdős-Rényi graphs leveraging graph limits and regularity lemma, providing new insights into rare event probabilities.

Findings

Large deviation principle for fixed p as n→∞

Application to triangle counts shows phase transitions

Uses graph limit theory and Szemerédi's regularity lemma

Abstract

What does an Erdos-Renyi 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 Lovasz and coauthors and Szemeredi'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.