Scaling limits of discrete snakes with stable branching

TL;DR

This paper investigates the scaling limits of discrete snakes derived from critical Bienaymé-Galton-Watson trees with stable offspring distributions, establishing conditions for convergence to the Brownian snake driven by stable Lévy trees.

Contribution

It provides a necessary and sufficient condition for the convergence of spatial trees with stable offspring distributions to the Brownian snake driven by stable Lévy trees.

Findings

Convergence criteria for spatial trees with stable offspring distributions.

Extension to cases with heavier tails.

Application to counting inversions in random permutations.

Abstract

We consider so-called discrete snakes obtained from size-conditioned critical Bienaym\'e-Galton-Watson trees by assigning to each node a random spatial position in such a way that the increments along each edge are i.i.d. When the offspring distribution belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$, we give a necessary and sufficient condition on the tail distribution of the spatial increments for this spatial tree to converge, in a functional sense, towards the Brownian snake driven by the $\alpha$-stable L\'evy tree. We also study the case of heavier tails, and apply our result to study the number of inversions of a uniformly random permutation indexed by the tree.