Unusually large components in near-critical Erd\H{o}s-R\'{e}nyi graphs via ballot theorems

TL;DR

This paper offers a new probabilistic proof for the size distribution of large components in near-critical Erdős-Rényi graphs, using ballot theorems instead of combinatorial formulas.

Contribution

It introduces a conceptual, adaptable proof method based on ballot theorems for analyzing large components in near-critical Erdős-Rényi graphs.

Findings

Provides asymptotic probability estimates for large components

Allows parameters to depend on the number of nodes

Uses ballot theorems for a more conceptual proof

Abstract

We consider the near-critical Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ and provide a new probabilistic proof of the fact that, when $p$ is of the form $p=p(n)=1/n+\lambda/n^{4/3}$ and $A$ is large, \[\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3}{8}+\frac{\lambda A^2}{2}-\frac{\lambda^2A}{2}}\] where $\mathcal{C}_{\max}$ is the largest connected component of the graph. Our result allows $A$ and $\lambda$ to depend on $n$. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erd\H{o}s-R\'{e}nyi graphs, together with analytic estimates.