The Spectrum and Eigenvectors of Random Graphs

The goal of this project is to understand the distribution of eigenvalues and eigenvectors of large random graphs. Random graph theory is an important topic that lies at the intersection of probability theory and graph theory. It can be regarded as the statistical approach to spectral graph theory, and it is particularly efficient in answering questions about the properties of typical graphs. From the application point of view, random graphs serve as effective models in the study of complex networks. We normally study a random graph through its adjacency matrix, which is simply a matrix whose entries are random variables, and thus, a random matrix. Random matrix theory (RMT) is a very active area of mathematics, whose origins lie in the work of Wigner in the fifties. A central topic in RMT is to study the behaviour of the individual eigenvalues, which we regard as "the microscopic eigenvalue statistics". The famous universality conjecture of random matrices asserts that the microscopic eigenvalue statistics depend only on the symmetry class of the matrix, instead of the law of the matrix elements. By the powerful three-step approach developed during the last decade, the universality conjecture has been established in great generality for the model of Wigner matrices. Unlike Wigner matrices, many important problems concerning random graphs are still unsolved, as the techniques and methods used for Winger matrices are often insufficient in the random graph setting. Regarding this issue, lots of new ideas have emerged in the last few years, which led to a number of promising results. In this project, we plan to continue generating new approaches, in the hope of answering important open questions concerning random graph models. In particular, we will study the spectral statistics of Erdős–Rényi graphs and random regular graphs. These are the two central random graph models and are of prominent interest. Our aim is to develop a systematic scheme to deeply promote the understanding of the universality phenomenon in random graphs. This project will enhance our knowledge of random graphs and random networks, thus making impacts on both mathematics and computer science.