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

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