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.