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Condensation in critical Cauchy Bienaymé-Galton-Watson trees

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RGMW06 - RGM follow up

We will be interested in the structure of large Bienaymé-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index α=1. In stark contrast to the case α∈(1,2], we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges. One of the main tools is a limit theorem for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called non-generic of parameter 3/2) and support the conjecture that faces in Le Gall & Miermont's 3/2-stable maps are self-avoiding. This is joint work with Loïc Richier.

This talk is part of the Isaac Newton Institute Seminar Series series.

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