Discrete snakes with globally centered displacements

TL;DR

This paper establishes scaling limits for discrete snakes on critical Bienaymé trees, showing convergence to Brownian and hairy snakes under various conditions, with applications to tree height processes and looptrees.

Contribution

It proves new convergence results for discrete snakes on Bienaymé trees, including uniform and heavy-tailed cases, and applies these to tree height and looptree analysis.

Findings

Convergence to Brownian snake under finite variance.

Uniform convergence with finite third moment.

Heavy-tailed displacements lead to hairy snake convergence.

Abstract

We prove a scaling limit for globally centered discrete snakes on size-conditioned critical Bienaym\'e trees. More specifically, under a global finite variance condition, we prove convergence in the sense of random finite-dimensional distributions of the head of the discrete snake (suitably rescaled) to the head of the Brownian snake driven by a Brownian excursion. When the third moment of the offspring distribution is finite, we further prove uniform functional convergence under a necessary tail condition on the displacements. We also consider displacement distributions with heavier tails, for which we instead obtain convergence to a variant of the hairy snake introduced by Janson and Marckert. We further give two applications of our main result. Firstly, we obtain a scaling limit for the difference between the height process and the {\L}ukasiewicz path of a size-conditioned critical Bienaym\'e tree. Secondly, we obtain a scaling limit for the difference between the height process of a size-conditioned critical Bienaym\'e tree and the height process of its associated looptree.