Scaling limits of discrete snakes with stable branching

Abstract

We consider so-called discrete snakes obtained from size-conditioned critical Bienaymé-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 Lévy 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.

arXiv commentMore figures, as promised, 8 in total; 29 pages

References in corpus (10)

• Random trees and applications
• Invariance principles for Galton-Watson trees conditioned on the number of leaves
• Scaling limits of random planar maps with a unique large face
• Invariance principles for random bipartite planar maps
• Local limits of conditioned Galton-Watson trees I: the infinite spine case
• A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees
• Scaling limits of random bipartite planar maps with a prescribed degree sequence
• Random walk on random infinite looptrees
• The lineage process in Galton--Watson trees and globally centered discrete snakes
• Inversions in split trees and conditional Galton--Watson trees

Cited by in corpus (3)

• Convergence of blanket times for sequences of random walks on critical random graphs
• Rotating random trees with Skorokhod's $M_1$ topology
• Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees