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，而且在科学中也随处可见。它们自然而然地出现在对一系列现象的建模中：进化生物学中的系谱，生态学中的种群传播和生存，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的范畴。当然，树是经典的图论对象。但是，更广泛地说，哪里有递归，哪里就有一棵树，所以分支结构出现在令人惊讶的各种数学背景中，就像几何学和算法复杂性的分析一样。在概率论中，这是最真实的。与本项目特别相关的是关于分支随机游动的极端粒子附近发生的一些最新发展。更准确地说，如果一个人在很长一段时间内考虑分枝的布朗运动，并观察右边最远的粒子，它在Aidekon等人中显示了出来。(2013)和Arguin等人的。(2013)，人们看到一个极限对象恰好是一个装饰的泊松点过程，这是一个泊松点过程，其中每个原子都被某个点度量的独立副本装饰。最近，许多人致力于了解这种装饰点措施，例如见Berestycki等人。(2022)，Brunet et al.(2020)，仅举几部最近的作品。第一个项目是基于物理学家Le等人最近的一项研究。(2022)关于装饰措施中粒子数量的波动。利用分枝布朗运动的倒向构造，我们将证实他们的预言，得到一个严格的中心极限定理，以及关于分枝布朗运动极点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上的马尔可夫链。更准确地说，有各种各样的自然动力。这里的建议是考虑非平面二叉树上的Aldous自下而上马尔可夫链，并给它们配备一个平面序。具体地说，奥尔多斯的马尔可夫链随机地取走一片叶子，并将它移动到一个也是随机均匀选择的新位置。在平面树树高函数的设置中，选择和去掉树叶具有自然意义。Marchal(2003)在建立随机高度函数到布朗游程的强收敛的研究中研究了在新位置的插入。Aldous(2000)和Schweinsberg(2002)建立了Aldous马尔可夫链的n^2阶松弛时间，并且可以预期对高度函数的马尔可夫链也是如此。这项工作的动机是Forman等人的工作。(2020,2022+)在Aldous(1999)猜想的连续统树空间上建立一个极限扩散过程。高度函数形式的附加平面结构允许在更丰富的连续游程函数设置中重塑这一问题，这提供了诸如随机偏微分方程组等附加技术。Zambotti(2017)的相关工作使用了这一技术来研究同一状态空间中不同马尔可夫链的极限。在这些环境中的任何一种情况下，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.