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

Ojeda, Gabriel Berzunza and Holmgren, Cecilia (2020) Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees. Bernoulli, 29 (4).

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Official URL: http://dx.doi.org/10.3150/22-bej1559

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 $n \rightarrow \infty$ 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 $\alpha \in (1,2]$. Our main results establish that, after rescaling, the fragmentation process of $\mathbf{t}_{n}$ converges as $n \rightarrow \infty$ to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an $\alpha$-stable L\'evy tree of index $\alpha \in (1,2]$. We further show that the latter can be constructed by considering the partitions of the unit interval induced by the normalized $\alpha$-stable L\'evy excursion with a deterministic drift studied by Miermont (2001). This extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.

Item Type:	Article
Additional Information:	30 pages, 5 figures
Uncontrolled Keywords:	math.PR, math.PR
Divisions:	Faculty of Science and Engineering > School of Physical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	15 Jan 2024 16:24
Last Modified:	15 Jan 2024 16:24
DOI:	10.3150/22-bej1559
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3177847