Discrete Snakes with Globally Centered Displacements
We consider certain size-conditioned critical Bienaymé trees, in which each vertex is endowed with a spatial location that is a random displacement away from their parent’s location. By construction, the positions along each vertex’s lineage form a random walk. It is convenient to encode the genealogy and spatial locations using a path-valued process called the discrete snake. We prove that under a global finite variance and a tail behaviour assumption on the displacements, any globally centered discrete snake on a Bienaymé tree whose offspring distribution is critical and admits a finite third moment has the Brownian snake driven by a normalised Brownian excursion as its scaling limit. Our proof relies on two perspectives of Bienaymé trees. To prove convergence of finite dimensional distributions we rely on a line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin. To prove tightness, we adapt a method used by Haas and Miermont in the context of height functions of Markov branching trees.

This is based on joint work in progress with Louigi Addario-Berry, Serte Donderwinkel, and Christina Goldschmidt.
Date: 21 October 2024, 14:00
Venue: Mathematical Institute, Woodstock Road OX2 6GG
Venue Details: L6
Speaker: Rivka Mitchell (University of Oxford)
Organising department: Department of Statistics
Part of: Probability seminar
Booking required?: Not required
Audience: Members of the University only
Editors: Christina Goldschmidt, James Martin