Topics in random graphs and 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.