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

TL;DR

This paper extends the understanding of fragmentation processes from conditioned Galton-Watson trees with stable offspring distributions, showing convergence to stable Lévy tree-based processes and connecting to stable Lévy excursions.

Contribution

It introduces a new invariance principle for fragmentation processes derived from critical Galton-Watson trees with stable offspring distributions, generalizing previous Brownian CRT results.

Findings

Fragmentation process converges to that of an $oldsymbol{ ext{α}}$-stable Lévy tree.

Constructs the limiting process via partitions induced by stable Lévy excursions.

Extends Bertoin's Brownian CRT fragmentation results to stable Lévy trees.

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.