## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 2301-2329.

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

#### 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.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 2301-2329.

**Dates**

Received: October 2017

Revised: June 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326446

**Digital Object Identifier**

doi:10.3150/18-BEJ1055

**Mathematical Reviews number (MathSciNet)**

MR3961249

**Zentralblatt MATH identifier**

07066258

**Keywords**

Continuum Random Tree cut-tree fragmentation at nodes Galton–Watson tree Gromov–Hausdorff–Prokhorov topology Invariance Principle $\mathbb{R}$-tree stable tree

#### Citation

He, Hui; Winkel, Matthias. Gromov–Hausdorff–Prokhorov convergence of vertex cut-trees of $n$-leaf Galton–Watson trees. Bernoulli 25 (2019), no. 3, 2301--2329. doi:10.3150/18-BEJ1055. https://projecteuclid.org/euclid.bj/1560326446