Convergence of the height process of supercritical Galton-Watson forests with an application to the configuration model in the critical window

TL;DR

This paper proves the joint convergence of key processes in supercritical Galton-Watson forests and applies these results to derive the metric space scaling limit of the configuration model with power-law degrees in the critical window.

Contribution

It establishes the convergence of the height process and Lukasiewicz path for supercritical Galton-Watson forests and applies this to the configuration model in the critical window.

Findings

Joint convergence of Lukasiewicz path and height process for supercritical forests

Limit of height processes as continuous state branching processes

Metric space scaling limit for configuration model with power-law degrees

Abstract

We show joint convergence of the Lukasiewicz path and height process for slightly supercritical Galton-Watson forests. This shows that the height processes for supercritical continuous state branching processes as constructed by Lambert (2002) are the limit under rescaling of their discrete counterparts. Unlike for (sub-)critical Galton-Watson forests, the height process does not encode the entire metric structure of a supercritical Galton-Watson forest. We demonstrate that this result is nonetheless useful, by applying it to the configuration model with an i.i.d. power-law degree sequence in the critical window, of which we obtain the metric space scaling limit in the product Gromov-Hausdorff-Prokhorov topology, which is of independent interest.