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

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.