Near critical asymptotics in the Frozen Erd\H{o}s-R\'enyi

TL;DR

This paper studies a modified Erdős-Rényi graph model where surplus components are slowed, leading to a 'frozen' critical window behavior, and analyzes the asymptotics of component sizes with surplus over time.

Contribution

It introduces a variant of Erdős-Rényi with slowed surplus components and characterizes the long-term asymptotics in the critical window.

Findings

Component sizes follow a 'frozen' Aldous' multiplicative coalescent in the critical window.

The total number of vertices in surplus components exhibits specific asymptotic behavior.

The phase transition is similar to the classical Erdős-Rényi model despite the modifications.

Abstract

We consider a variant of the classical Erd\H{o}s-R\'enyi random graph, where components with surplus are slowed down to prevent the apparition of complex components. The sizes of the components of this process undergo a similar phase transition to that of the classical model, and in the critical window the scaling limit of the sizes of the components is a "frozen" version of Aldous' multiplicative coalescent [2]. The aim of this article is to describe the long time asymptotics in the critical window for the total number of vertices which belong to a component with surplus.