The standard augmented multiplicative coalescent revisited

TL;DR

This paper revisits the standard augmented multiplicative coalescent process, providing a simpler, more direct approach to its scaling limit, which models the evolution of component sizes and surplus edges in a continuous-time Erdős-Rényi graph.

Contribution

It introduces a streamlined method using the simultaneous breadth-first walk to analyze the scaling limit of the SAMC, facilitating potential generalizations.

Findings

Simplified proof of the SAMC scaling limit

Use of the simultaneous breadth-first walk method

Potential for extending to non-standard models

Abstract

The Erd\H{o}s-R\'enyi random graph is the fundamental random graph model. In this paper we consider its continuous-time version, where multi-edges and self-loops are also allowed. It is well-known that the sizes of its connected components evolve according to the multiplicative coalescent dynamics. Moreover, with the additional information on the number of surplus edges, the resulting process follows the augmented multiplicative coalescent dynamic, constructed by Bhamidi, Budhiraja and Wang in 2014. The same authors exhibit the scaling limit, which can be specified in terms of the infinite vector of excursions (in particular their lengths, and the areas enclosed by the excursion curves) above past infima of a reflected Brownian motion with linear infinitesimal drift. We use some recent results, using a graph exploration process called the simultaneous breadth-first walk, to study the same scaling limit, called the standard augmented multiplicative coalescent (SAMC). We present a self-contained, simpler and more direct approach than that of any previous construction of the SAMC. Furthermore, we believe that the method described here is convenient for generalizations, one of which would be the study of general non-standard eternal augmented multiplicative coalescents.