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

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 our prior work 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.

Automated summary

In this paper, we establish the global lower mass-bound property of the largest connected components in the critical window for the configuration model with an infinite third moment in the degree distribution. We extend the scaling limit results of the critical percolation clusters, viewed as measured metric spaces, to the stronger Gromov-Hausdorff-Prokhorov topology under slightly stronger assumptions on the degree distribution. Our result implies the distributional convergence of global functionals such as the diameter of the largest critical components and provides a sufficient condition for the compactness of the random metric spaces in the heavy-tailed regime.