
Title
Discrete Snakes with Globally Centered Displacements
Abstract
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.
Please note that the seminar will take place in person in room 144 of Huxley Building.