Global lower mass-bound for critical configuration models in the heavy-tailed regime

TL;DR

This paper proves the global lower mass-bound property for critical configuration model components with heavy-tailed degree distributions, extending previous scaling limit results to stronger topologies and implications for compactness and diameter convergence.

Contribution

It extends the scaling limit analysis of critical configuration models to the Gromov-Hausdorff-Prokhorov topology under heavy-tailed degrees, establishing new compactness and convergence results.

Findings

Proves global lower mass-bound for critical components with infinite third moment

Extends scaling limits to Gromov-Hausdorff-Prokhorhov topology

Provides conditions for compactness of scaling limits

Abstract

We establish the global lower mass-bound property for the largest connected components in the critical window for the configuration model when the degree distribution has an infinite third moment. The scaling limit of the critical percolation clusters, viewed as measured metric spaces, was established in [7] with respect to the Gromov-weak topology. Our result extends those scaling limit results to the stronger Gromov-Hausdorff-Prokhorov topology under slightly stronger assumptions on the degree distribution. This implies the distributional convergence of global functionals such as the diameter of the largest critical components. Further, our result gives a sufficient condition for compactness of the random metric spaces that arise as scaling limits of critical clusters in the heavy-tailed regime.