# Gromov-Hausdorff-Prokhorov convergence of vertex cut-trees of n-leaf Galton-Watson trees

## TL;DR

This paper proves that the vertex cut-trees of conditioned Galton-Watson trees converge to the Brownian CRT and its cut-tree in the Gromov-Hausdorff-Prokhorov sense, extending previous convergence results.

## Contribution

It establishes joint Gromov-Hausdorff-Prokhorov convergence for vertex cut-trees of Galton-Watson trees conditioned on leaves, in the finite variance case, and discusses extensions to infinite variance.

## Key findings

- Convergence of vertex cut-trees to Brownian CRT in finite variance case
- Methods extend to infinite variance case
- Open problem for strengthened convergence in conditioned models

## Abstract

In this paper we study the vertex cut-trees of Galton-Watson trees conditioned to have $n$ leaves. This notion is a slight variation of Dieuleveut's vertex cut-tree of Galton-Watson trees conditioned to have $n$ vertices. Our main result is a joint Gromov-Hausdorff-Prokhorov convergence in the finite variance case of the Galton-Watson tree and its vertex cut-tree to Bertoin and Miermont's joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut's and Bertoin and Miermont's Gromov-Prokhorov convergence to Gromov-Hausdorff-Prokhorov remains open for their models conditioned to have $n$ vertices.