Topics in random graphs and discrete snakes

网络是表示系统中大量组件相互作用的一种简单而有效的方法。随着科学、技术和社会研究的应用，它们在今天的相关性是至关重要的。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源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，但它也包括对概率学家和组合学家来说有趣的数学问题。近年来，人们对随机图的尺度极限进行了大量的研究。这项研究的核心是连续随机树（CRT），由Aldous在20世纪90年代初提出。简而言之，CRT是一个连续的树状结构，它表现为特定大小条件的bienaym<s:1>树的缩放极限。自Aldous的工作以来，许多人利用CRT来研究各种离散结构的缩放极限。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，重点关注随机树上离散蛇的缩放限制。非正式地说，这是一个分支结构，除了有一个谱系，每个节点都有一个相关的轨迹。在极限情况下，蛇形结构结合了CRT的连续结构和由马尔可夫过程控制的独立空间运动。离散蛇的应用也是多种多样的。正如勒加尔所讨论的，蛇与偏微分方程有联系。另一方面，蛇通常被用来研究随机平面图的收敛性。例如，参见Miermont和Le Gall关于均匀随机平面四边形的缩放极限的著作，以及Addario-Berry和Albenque关于随机简单三角形和球体四边形的缩放极限的著作。离散蛇的收敛性已经在许多情况下得到了研究。Janson和markert共同研究了离散蛇在具有有限指数矩的子代分布的大小条件bienaym<s:1>树上的收敛性，其中位移为i.i.d，平均值为零，并满足一定的矩条件。Marzouk后来建立了类似离散蛇在大小条件临界bienaym<s:1>树上的收敛结果，其后代分布属于稳定定律的吸引域。也许与这个项目最相关的是，在2008年，markkert证明了以全球为中心的蛇在关键的bienaym<e:1>树上的收敛性，这些树的后代分布有有限的支持。这个结果特别强大，因为它允许人们考虑更广泛的位移分布，包括确定性位移。当使用蛇来理解随机图的属性时，这可以作为一个强大的工具。在这个项目中，一个具体的探索途径是建立在markert的工作基础上，确定某些全球中心蛇在具有无界支持的后代分布的临界bienaym<s:1>树上的缩放极限。作为第一步，我们打算将注意力集中在泊松(1)后代分布树上的离散蛇。该项目属于EPSRC数学分析、统计与应用概率、逻辑学与组合学研究领域，由Christina Goldschmidt教授指导。

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.