Embedded Tree Structures and Eigenvalue Statistics of Genus Zero One-Face Maps

Abstract

Using numerical simulations and combinatorics, this dissertation focuses on connections between random matrix theory and graph theory.We examine the adjacency matrices of three-regular graphs representing one-face maps. Numerical studies have revealed that the limiting eigenvalue statistics of these matrices are the same as those of much larger, and more widely studied classes of random matrices. In particular, the eigenvalue density is described by the McKay density formula, and the distribution of scaled eigenvalue spacings appears to be that of the Gaussian Orthogonal Ensemble (GOE).A natural question is whether the eigenvalue statistics depend on the genus of the underlying map. We present an algorithm for generating random three-regular graphs representing genus zero one-face maps. Our numerical studies of these three-regular graphs have revealed that their eigenvalue statistics are strikingly different from those of three-regular graphs representing maps of higher genus. While our results indicate that there is a limiting eigenvalue density formula in the genus zero case, it is not described by any established density function. Furthermore, the scaled eigenvalue spacings appear to be described by the exponential distribution function, not the GOE spacing distribution.The embedded graph of a genus zero one-face map is a planar tree, and there is a correlation between its vertices and the primitive cycles of the associated three-regular graph. The second half of this dissertation examines the structure of these embedded planar trees. In particular, we show how the Dyck path representation can be used to recast questions about the probabilistic structure of random planar trees into straightforward counting problems. Using this Dyck path approach, we find:1. the expected number of degree k vertices adjacent to j degree d vertices in a random planar tree, 2. the structure of the planar tree's adjacency matrix under a natural labeling of the vertices, and 3. an explanation for the existence of eigenvalues with multiplicity greater than one in the tree's spectrum.