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

Serte Donderwinkel

Published 2021-05-25Version 1

We show joint convergence of the {\L}ukasiewicz 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 counterpart. 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 can nevertheless be used, 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 Gromov-Hausdorff-Prokhorov product topology, which is of individual interest.