2020/10/15 by Gabriel Berzunza Ojeda, Ojeda, Gabriel Berzunza, Cecilia Holmgren +1 · 1 citation
Decision Sciences · Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2010.07880
openalex publication_date 2020/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 tn conditioned on having n vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index α∈ (1,2]. Our main results establish 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 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.