The phase transition in random graphs - a simple proof

TL;DR

This paper presents a simplified proof of the classical phase transition in Erdős-Rényi random graphs, demonstrating the emergence of a giant component and linear paths in the supercritical regime.

Contribution

It provides a straightforward proof of the phase transition in G(n,p), highlighting the existence of linear-length paths in the supercritical phase and extending the technique to other models.

Findings

In the supercritical regime, G(n,p) contains a linear-length path with high probability.

The proof simplifies understanding of the phase transition phenomenon.

Applications extend to other random graph models and positional games.

Abstract

The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any \epsilon>0 and p=(1-\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while for p=(1+\epsilon)/n, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime p=(1+\epsilon)/n, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games.