The probability of unusually large components in the near-critical Erd\H{o}s-R\'enyi graph

TL;DR

This paper provides detailed asymptotic probabilities for unusually large components in near-critical Erdős-Rényi graphs, extending previous work and including size distribution of specific vertex components.

Contribution

It extends existing results by deriving asymptotics for large component probabilities in near-critical Erdős-Rényi graphs, allowing for variable parameters.

Findings

Asymptotics for probability of large components of size $an^{2/3}$

Distribution of component sizes containing a specific vertex

Results valid for $p$ around $1/n$

Abstract

The largest components of the critical Erd\H{o}s-R\'enyi graph, $G(n,p)$ with $p=1/n$, have size of order $n^{2/3}$ with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of size $an^{2/3}$ for large $a$. Our results, which extend work of Pittel, allow $a$ to depend upon $n$ and also hold for a range of values of $p$ around $1/n$. We also provide asymptotics for the distribution of the size of the component containing a particular vertex.