Events Archive

Benedikt Stufler (TU Wien) Limits of Random Trees: A Probabilistic Perspective on Combinatorial Tree Models In this talk, we explore the asymptotic behaviour of random trees arising from various combinatorial models, with a focus on probabilistic techniques for analysing their limiting structures. We will discuss key results in the convergence of random trees to continuous limits, such as Aldous' Brownian tree and related limit objects. Christina Goldschmidt (Univ. Oxford) Trees and snakes Consider the following branching random walk model: individuals reproduce according to a branching process which is started from a single individual who is located at the origin. The children of a vertex receive random displacements away from the spatial location of the parent. The displacements away from a particular vertex may be dependent, with a distribution depending on the number of children. However, the set of displacements of a vertex’s children are independent of the sets of displacements associated with the children of other vertices. We are interested in the setting where the genealogical tree is conditioned to have precisely n vertices and the offspring distribution is critical and of finite variance, so that the tree converges on rescaling distances by n^{-1/2} to Aldous’ Brownian continuum random tree. We also assume that the spatial displacements satisfy natural centring and finite variance conditions, so that, along a particular lineage, we observe something like a centred finite variance random walk. We investigate conditions under which the whole object then converges in distribution (in a suitable sense) to a Brownian motion indexed by the Brownian continuum random tree. (In order to do this, we actually make use of a standard tool known as a discrete snake, and prove convergence on rescaling to Le Gall’s Brownian snake driven by a Brownian excursion.) Our results improve on earlier theorems of various authors, including Janson and Marckert for the case where the displacements are independent of the offspring numbers, and Marckert for the globally centred, global finite variance case restricted to bounded offspring distributions. Our proof of the convergence of the finite dimensional distributions makes essential use of a discrete line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin; the tightness proof adapts a method deployed by Haas and Miermont in the context of Markov branching trees. This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.