Some contributions to the study of branching structures and trees

树和分支结构不仅在数学中无处不在，而且在科学中也无处不在。它们自然地出现在大量现象的建模中：进化生物学中的谱系，生态学中种群的传播和生存，计算机科学中的数据结构，等等。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，所以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。这在概率论中是最真实的。与本项目特别相关的是关于分支随机漫步的极端粒子附近发生的情况的一些最新进展。更准确地说，如果考虑大时间的分支布朗运动，并向右看最远的粒子附近，Aidekon等人（2013）和Arguin等人（2013）表明，人们看到的极限对象恰好是一个修饰泊松点过程，即泊松点过程中每个原子都被某个点测量的独立副本修饰。最近，很多工作都致力于理解这一装饰点测量，例如Berestycki等人（2022），Brunet等人（2020）引用了最近的一些作品。第一个项目是基于物理学家Le等人（2022）最近的一项研究，该研究涉及装饰措施中粒子数量的波动。通过从分支布朗运动的尖端看分支布朗运动的反向构造，我们将证实他们的预测，并得到一个严格的中心极限定理以及关于分支布朗运动的极值过程的几个有趣的结果。第二个项目是研究给定长度的高度函数的马尔可夫链。更准确地说，有各种各样的自然动态。这里的建议是考虑非平面二叉树上的Aldous的上下马尔可夫链，并赋予它们一个平面阶。具体来说，Aldous's Markov链均匀随机地取一个叶子，并将它移动到一个同样均匀随机选择的新位置。在平面树的高度函数设置中，选择和移除叶子具有天然的意义。Marchal（2003）在一项研究中研究了新位置的插入，该研究建立了随机高度函数对布朗偏移的强收敛性。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。这项工作的一个动机是Forman等人（2020,2022+）在Aldous（1999）猜想的连续统树空间上建立了一个极限扩散过程。高度函数形式的附加平面结构允许在更丰富的连续偏移函数设置中重新定义这个问题，这提供了额外的技术，如随机偏微分方程。Zambotti（2017）的相关工作使用该技术研究了同一状态空间中不同马尔可夫链的极限。在任何这些设置中，限制对象被期望表现出普遍性。

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.