Topics in random graphs and discrete snakes

随机图和离散蛇中的主题

• 批准号: 2594689
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2594689
• 关键词: Topics; random; graphs; discrete; snakes

Networks are a simple yet effective way of representing the interaction of a large number of components in a system. With applications in science, technology, and social studies, their relevance today is paramount. More specifically, as large datasets become increasingly commonplace, various disciplines face the challenge of interpreting data originating from large networks. In mathematics, networks are modelled by graphs. The topic of random graphs is an active area of research, and while the study of random graphs has high cross disciplinary impact, it also comprises of interesting mathematical problems for probabilists and combinatorialists alike. In recent years, much work has been conducted on the topic of the scaling limits of random graphs. Central to this study is the Continuum Random Tree (CRT), introduced by Aldous in the early 1990s. To be brief, the CRT is a continuous tree-like structure which appears as the scaling limit of certain size-conditioned Bienaymé trees. Since the work of Aldous, many have drawn on the CRT to study the scaling limits of various discrete structures. This project aims to do the same, and build on this theory with a focus on the scaling limits of discrete snakes on random trees. Informally, this is branching structure where in addition to having a genealogy, each node has an associated trajectory. In the limit, the snake structure combines the continuous structure of the CRT with independent spatial motions governed by a Markov process. The applications of discrete snakes are again varied. As discussed by Le Gall, snakes have connections to partial differential equations. On the other hand, snakes are often employed to study the convergence of random planar maps. See for example the works of Miermont, and Le Gall, on the scaling limits of uniform random plane quadrangulations, as well as that by Addario-Berry and Albenque concerning the scaling limits of random simple triangulations and quadrangulations of the sphere. The convergence of discrete snakes has been studied in many settings. Together, Janson and Marckert studied the convergence of discrete snakes on size conditioned Bienaymé trees with offspring distribution having finite exponential moments, where the displacements are i.i.d, mean zero, and satisfy certain moment conditions. Marzouk later established convergence results for similar discrete snakes on size conditioned critical Bienaymé trees whose offspring distribution belongs to the domain of attraction of a stable law. Of perhaps highest relevance to this project, in 2008, Marckert proved convergence of globally centred snakes on critical Bienaymé trees whose offspring distribution have bounded support. This result is particularly powerful, as it allows one to consider a wider class of displacement distributions, including deterministic displacements. This can serve as a strong tool when using snakes to understand properties of random graphs. One specific avenue for exploration in this project is to build on the work of Marckert to determine the scaling limit of certain globally centred snakes on critical Bienaymé trees with offspring distribution having unbounded support. As a first step, we intend to focus our attention to discrete snakes on trees with Poisson(1) offspring distribution. This project falls within the EPSRC Mathematical Analysis, Statistics and Applied Probability, and Logic and Combinatorics research areas, and is supervised by Professor Christina Goldschmidt.

网络是一种简单而有效的方法，可以表示系统中大量组件的交互。随着科学，技术和社会研究的应用，它们的相关性今天是至关重要的。更具体地说，随着大型数据集变得越来越普遍，各种学科都面临着解释来自大型网络的数据的挑战。在数学中，网络是用图来建模的。随机图的主题是一个活跃的研究领域，虽然随机图的研究具有很高的跨学科影响，但它也包括概率学家和组合学家的有趣数学问题。近年来，人们对随机图的标度极限进行了大量的研究。这项研究的核心是连续统随机树（CRT），由Aldous在20世纪90年代初提出。简单地说，CRT是一个连续的树状结构，它表现为某些尺寸条件下的Bienayme树的标度极限。自从Aldous的工作以来，许多人利用CRT来研究各种离散结构的标度极限。这个项目的目标是做同样的事情，并建立在这个理论的基础上，重点是随机树上离散蛇的缩放限制。非正式地说，这是一种分支结构，其中除了具有家谱之外，每个节点都具有相关联的轨迹。在极限情况下，蛇形结构将CRT的连续结构与由马尔可夫过程控制的独立空间运动相结合。离散蛇的应用也是多种多样的。正如Le Gall所讨论的，蛇与偏微分方程有联系。另一方面，蛇经常被用来研究随机平面映射的收敛性。例如，见作品的米尔蒙，和勒加勒，对缩放限制的均匀随机平面四边形，以及由Addario-Berry和Albenque关于缩放限制的随机简单三角形和四边形的领域。离散蛇的收敛性已经在许多场合得到了研究。Janson和Marckert一起研究了离散蛇在具有有限指数矩的后代分布的尺寸条件Bienaymé树上的收敛性，其中位移是i.i.d，平均为零，并满足一定的矩条件。Marzouk后来建立了类似的离散蛇收敛结果的大小条件临界Bienaymé树的后代分布属于域的吸引力稳定的法律。也许与该项目最相关的是，2008年，Marckert证明了以全球为中心的蛇在关键Bienaymé树上的收敛，这些树的后代分布具有有限的支持度。这个结果特别强大，因为它允许考虑更广泛的位移分布，包括确定性位移。这可以作为一个强大的工具，使用蛇来理解随机图的属性。在这个项目中探索的一个具体途径是建立在Marckert的工作，以确定某些全球为中心的蛇的缩放限制关键Bienayme树的后代分布具有无限的支持。作为第一步，我们打算把我们的注意力集中到离散蛇的树泊松（1）的后代分布。该项目福尔斯EPSRC数学分析，统计和应用概率，逻辑和组合学研究领域，并由Christina Goldschmidt教授监督。