Excursion Theory, Galton Watson Trees and their Scaling Limits

In this talk we aim to introduce a recent perspective in probability theory that views random trees as random excursions with additional data. This perspective is particular suited to the study of the scaling limit of tree-valued random processes. Excursion theory is a useful and relatively elementary tool allowing one to derive rather explicit information about the local and global geometry of the resultant continuum trees which in turn can be used to derive information about large random trees. We illustrate these ideas in the context of the Brownian continuum random tree, the scaling limit of critical Galton-Watson trees and a structure that arises naturally in various contexts in physics; in particular the Brownian continuum random tree is a pathological model of quantum spacetime. Despite the fundamentally mathematical nature of the talk, the aim is to keep the presentation essentially heuristic emphasising key intuitions over rigorous proof. The content itself should be relevant to biologists interested in the theory of branching processes or coalescent theory.