The vertex-cut-tree of Galton–Watson trees converging to a stable tree

Abstract

We consider a fragmentation of discrete trees where the internal vertices are deleted independently at a rate proportional to their degree. Informally, the associated cut-tree represents the genealogy of the nested connected components created by this process. We essentially work in the setting of Galton–Watson trees with offspring distribution belonging to the domain of attraction of a stable law of index $\alpha\in(1,2)$. Our main result is that, for a sequence of such trees $\mathcal{T}_{n}$ conditioned to have size $n$, the corresponding rescaled cut-trees converge in distribution to the stable tree of index $\alpha$, in the sense induced by the Gromov–Prokhorov topology. This gives an analogue of a result obtained by Bertoin and Miermont in the case of Galton–Watson trees with finite variance.