Fragmentation Processes Derived from Conditioned Stable Galton-Watson Trees

Abstract

We study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree tn conditioned on having n vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index α ∈ (1, 2]. This fragmentation process is analogous to that introduced in the works of Aldous, Evans and Pitman (1998), who considered the case of Cayley trees. Our main result establishes that, after rescaling, the fragmentation process of tn converges as n → ∞ to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an α-stable Lévy tree of index α ∈ (1, 2]. We further establish 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). In particular, this extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT. 2012 ACM Subject Classification Mathematics of computing → Probabilistic algorithms