Clay Mathematics Institute

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A line-breaking construction of the stable trees

Christina Goldschmidt (University of Oxford)

Abstract:  Consider a critical Galton-Watson tree whose offspring distribution lies in the domain of attraction of a stable law of parameter α ∈ (1,2], conditioned to have total progeny n.  The stable tree with parameter α ∈ (1,2] is the scaling limit of such a tree, where the α = 2 case is Aldous' Brownian continuum random tree. In this talk, I will discuss a new, simple construction of the α-stable tree for α ∈ (1,2]. We obtain it as the closure of an increasing sequence of R-trees built by gluing together line-segments one-by-one. The lengths of these line-segments are related to the the increments of an increasing R+-valued Markov chain. For α = 2, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.  This is joint work with Bénédicte Haas (Paris-Dauphine and ENS)