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Fragmentation Process derived from $\alpha$-stable Galton-Watson trees
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 talk, we will discuss the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree $t_n$ conditioned on having $n$ vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index $\alpha \in (1,2]$. The main result establishes that, after rescaling, the fragmentation process of $t_n$ converges, as grows to infinity, to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an $\alpha$-stable L\‘evy tree. We will also explain how one can construct the latter by considering the partitions of the unit interval induced by the normalized $\alpha$-stable L\‘evy excursion with a deterministic drift. In particular, the above extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.
The approach uses the well-known Prim’s algorithm (or Prim-Jarník algorithm) to define a consistent exploration process that encodes the fragmentation process of $t_n$. We will discuss the key ideas of the proof.
Date:
16 June 2025, 14:00
Venue:
Mathematical Institute, Woodstock Road OX2 6GG
Venue Details:
L4
Speaker:
Gabriel Berzunza Ojeda (University of Liverpool)
Organising department:
Department of Statistics
Part of:
Probability seminar
Booking required?:
Not required
Audience:
Members of the University only
Editors:
Christina Goldschmidt,
Julien Berestycki
