Invariance principle for fragmentation processes derived from conditioned stable Galton–Watson trees

Abstract

Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on n labelled vertices. In particular, they showed that, after proper rescaling, the above fragmentation process converges as $n\to \mathrm{\infty }$ to the fragmentation process of the Brownian CRT obtained by cutting-down the Brownian CRT along its skeleton in a Poisson manner.

In this work, we continue the above investigation and study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton–Watson tree ${\mathbf{t}}_n$ conditioned on having n vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index $\mathit{\alpha }\in (1,2]$. Our main results establish that, after rescaling, the fragmentation process of ${\mathbf{t}}_n$ converges as $n\to \mathrm{\infty }$ to the fragmentation process obtained by cutting-down at a rate proportional to the length measure on the skeleton of an α-stable Lévy tree. We further show that the latter can be constructed by considering the partitions of the unit interval induced by the normalized α-stable Lévy excursion with a deterministic drift studied by Miermont (2001). This extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.