Some contributions to the study of branching structures and trees

树和分支结构不仅在数学中，而且在科学中无处不在。它们自然地出现在对大量现象的建模中：进化生物学中的系谱，生态学中种群的传播和生存，计算机科学中的数据结构，仅举几例。然而，它们的重要性远远超出了建模。当然，树是经典的图论对象。但是，更广泛地说，哪里有递归，哪里就有树，因此分支结构出现在各种各样的数学背景中，就像几何和算法复杂性分析一样。这在概率论中是最正确的，与本项目特别相关的是关于分支随机游动的极值粒子附近发生的事情的一些最新进展。更准确地说，如果我们考虑一个分支布朗运动在很大的时间，并在最右边的粒子附近观察，Aidekon et al.（2013）和Arguin et al.（2013）表明，我们看到一个极限对象，它恰好是一个装饰的泊松点过程，也就是每个原子都被某个点测度的独立副本装饰的泊松点过程。最近，许多工作致力于理解这种装饰点度量，例如Berestycki et al.（2022），Brunet et al.（2020）引用了一些最近的工作。第一个项目是基于物理学家Le et al.（2022）关于装饰测度中粒子数波动的最新研究。通过从分支布朗运动的顶点向后构造分支布朗运动，我们将证实他们的预言，并得到一个严格的中心极限定理以及关于分支布朗运动极值点过程的几个有趣的结果.第二个项目是研究马尔可夫链的高度函数的一个给定的长度。更确切地说，有各种各样的自然动态。这里的建议是考虑Aldous的非平面二叉树上的自下而上的马尔可夫链，并为它们配备一个平面顺序。具体地说，奥尔德斯的马尔可夫链随机均匀地取一片叶子，并将其移动到一个同样随机均匀选择的新位置。在平面树高度函数的设置中，选择和去除叶子具有自然的意义。Marchal（2003）在一项研究中研究了在新位置的插入，该研究建立了随机高度函数到布朗偏移的强收敛。Aldous（2000）和Schweinsberg（2002）为Aldous的马尔可夫链建立了n^2阶的松弛时间，可以预期高度函数上的马尔可夫链也是如此。这项工作的动机是Forman et al.（2020，2022+）在Aldous（1999）提出的连续树空间上建立了一个极限扩散过程。高度函数形式主义的附加平面结构允许在连续偏移函数的更丰富设置中重铸这个问题，这提供了诸如随机偏微分方程的附加技术。Zambotti（2017）的相关工作使用这种技术来研究同一状态空间中不同马尔可夫链的极限。在任何一种情况下，限制对象都应该表现出普遍性。

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.