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 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 conditioned on having n vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index . Our main results establish that, after rescaling, the fragmentation process of converges as 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.
Funding Statement
This work was supported by the Knut and Alice Wallenberg Foundation, a grant from the Swedish Research Council and The Swedish Foundations’ starting grant from Ragnar Söderbergs Foundation.
Citation
Gabriel Berzunza Ojeda. Cecilia Holmgren. "Invariance principle for fragmentation processes derived from conditioned stable Galton–Watson trees." Bernoulli 29 (4) 2745 - 2770, November 2023. https://doi.org/10.3150/22-BEJ1559
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