Asymptotic normality of the size of the giant component via a random walk

TL;DR

This paper presents a new, simplified proof for the asymptotic distribution of the giant component size in Erdős–Rényi graphs, improving upon previous martingale-based methods.

Contribution

It introduces a novel martingale approach that yields sharper results on the giant component's distribution in random graphs.

Findings

Asymptotic normality of the giant component size established

New martingale proof simplifies previous methods

Sharper distribution results compared to earlier work

Abstract

In this paper we give a simple new proof of a result of Pittel and Wormald concerning the asymptotic value and (suitably rescaled) limiting distribution of the number of vertices in the giant component of $G(n,p)$ above the scaling window of the phase transition. Nachmias and Peres used martingale arguments to study Karp's exploration process, obtaining a simple proof of a weak form of this result. We use slightly different martingale arguments to obtain a much sharper result with little extra work.