25/06/2025, 17:00 — 18:00 — Online
Minmin Wang, University of Sussex
Exchangeability in continuum random trees
De Fenetti’s Theorem states that all N-indexed exchangeable sequences of real-valued random variables are mixings of i.i.d sequences. For real-valued random processes with exchangeable increments on [0, 1], Kallenberg’s 1973 result provides a complete characterisation of these processes via another mixing relationship. Continuum random trees are random tree-like metric spaces that arise naturally as scaling limits of various models of discrete random trees. In this talk, we will focus in particular on two subclasses of continuum random trees: the so-called stable trees and inhomogeneous continuum random trees. An analogue of Kallenberg’s Theorem for continuum random trees first appeared as a claim in a 2004 paper by Aldous, Miermont and Pitman. They suggested that, in much the same way that a stable bridge process on [0, 1] is a mixing of certain extremal exchangeable processes, stable trees are mixings of inhomogeneous continuum random trees. We present an outline of a rigorous argument supporting this claim, based on a novel construction that applies to both classes of trees. We will also briefly discuss some implications of this result on critical random graphs.
