Probability and combinatorics

• Date:
• Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
• Lecturer: Gabriel Berzunza-Ojeda, University of Liverpool
• Contact person: Paul Thévenin

Title: Fragmentation Process derived from $\alpha$-stable Galton-Watson trees

The seminar will take place at Ångström Laboratory, but it will also be possible to attend on Zoom. The passcode is the first six digits of pi after the decimal point.

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 -> \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 n -> \infty, to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an \alpha-stable Lévy 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 Lévy 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 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.

Joint work with Cecilia Holmgren (Uppsala University).