A note on the scaling limits of contour functions of Galton-Watson trees

TL;DR

This paper proves that the contour functions of truncated super-critical Galton-Watson trees converge weakly to the distributions of super-critical Lévy trees constructed via martingale transformations, linking discrete and continuous tree models.

Contribution

It establishes the weak convergence of contour functions of truncated super-critical Galton-Watson trees to Lévy tree distributions, extending existing convergence results to super-critical cases.

Findings

Contour functions of truncated super-critical Galton-Watson trees converge weakly.

The convergence links discrete Galton-Watson trees to continuous Lévy trees.

Extends known results from subcritical to super-critical tree models.

Abstract

Recently, Abraham and Delmas constructed the distributions of super-critical L\'evy trees truncated at a fixed height by connecting super-critical L\'evy trees to (sub)critical L\'evy trees via a martingale transformation. A similar relationship also holds for discrete Galton-Watson trees. In this work, using the existing works on the convergence of contour functions of (sub)critical trees, we prove that the contour functions of truncated super-critical Galton-Watson trees converge weakly to the distributions constructed by Abraham and Delmas.