Critical window for the configuration model: finite third moment degrees

TL;DR

This paper analyzes the critical behavior of the configuration model with finite third moment degrees, showing universal scaling limits for component sizes and their evolution through the critical window, extending known results from Erdős-Rényi graphs.

Contribution

It establishes that finite third moment degrees suffice for universal scaling limits in the critical configuration model, including component size distributions and evolution via percolation.

Findings

Component sizes scale as n^{2/3} at criticality.

Rescaled component sizes converge to inhomogeneous Brownian excursion lengths.

Percolation cluster sizes evolve as a multiplicative coalescent process.

Abstract

We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erd\H{o}s-R\'enyi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.