A simple path to component sizes in critical random graphs

TL;DR

This paper presents a unified and robust methodology using martingale and random walk techniques to derive bounds on the size of the largest component in critical random graphs, simplifying and unifying existing proofs.

Contribution

It introduces a common framework with easy conditions to obtain bounds on component sizes at criticality, consolidating various proofs into a single approach.

Findings

Unified methodology for component size bounds

Conditions that simplify derivations

Applicable to multiple random graph models

Abstract

We describe a robust methodology, based on the martingale argument of Nachmias and Peres and random walk estimates, to obtain simple upper and lower bounds on the size of a maximal component in several random graphs \textit{at criticality}. Even though the main result is not new, we believe the the material presented here is interesting because it unifies several proofs found in the literature into a common framework. More specifically, we give easy-to-check conditions that, when satisfied, allow an immediate derivation of the above mentioned bounds.