Some contributions to the study of branching structures and trees

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Trees and branching structures are ubiquitous not only in mathematics but across the sciences. They appear naturally in modelling a huge range of phenomena: genealogies in evolutionary biology, spread and survival of populations in ecology, data structures in computer science, to name but a few. However, their importance goes well beyond modelling. Of course, trees are classical graph-theoretic objects. But, more broadly, wherever there is recursion, there is a tree, and so branching structures turn up in an astonishing variety of mathematical contexts as disparate as geometry and the analysis of algorithmic complexity. Nowhere is this more true than in probability theory.Of particular relevance for the present project are some recent developments concerning what happen near the extremal particles of branching random walks. More precisely, if one considers a branching Brownian motion at large times and looks near the furthest particle to the right, it was shown in Aidekon et al. (2013) and in Arguin et al. (2013) that one sees a limit object which happens to be a decorated Poisson point process, that is a Poisson point process where each atom has been decorated by an independent copy of a certain point measure. More recently, a lot of work has been devoted to understanding this decoration point measure, see for instance Berestycki et al. (2022), Brunet et al. (2020) to cite just a few recent works. A first project is based on a recent study by physicists Le et al. (2022) concerning the fluctuations of the number of particles in the decoration measure. By using a backward construction of the branching Brownian motion seen from its tip, we will confirm their predictions and obtain a rigorous central limit theorem as well as several additional interesting results concerning the extremal point process of the branching Brownian motion. A second project is to study a Markov chain on height functions of a given length. More precisely, there are a variety of natural dynamics. The suggestion here is to consider Aldous's down-up Markov chain on non-planar binary trees and to equip them with a planar order. Specifically, Aldous's Markov chain takes a leaf uniformly at random and moves it to a new position also chosen uniformly at random. Choosing and removing a leaf has a natural meaning in the setting of height functions of planar trees. The insertion in a new position has been studied by Marchal (2003) in a study establishing a strong convergence of random height functions to a Brownian excursion. Aldous (2000) and Schweinsberg (2002) established relaxation times of order n^2 for Aldous's Markov chain, and it may be expected that the same holds for the Markov chain on height functions. A motivation for this work is work by Forman et al. (2020, 2022+) to establish a limiting diffusion process on a space of continuum trees conjectured by Aldous (1999). The additional planar structure of the height function formalism allows to recast this problem in the richer setting of continuous excursion functions, which offers additional techniques such as stochastic partial differential equations. Related work by Zambotti (2017) used this technique to study the limit of a different Markov chain in the same state space. In any of these settings, the limiting object is expected to exhibit universality.