Bernoulli

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

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

Hui He and Matthias Winkel

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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


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