Eigenvalue Densities for the Hermitian Two-Matrix Model and Connections to Hurwitz Numbers

Abstract

This dissertation investigates the limiting distribution of eigenvalues of pairs of matrices (M_1,M_2) belonging to the Hermitian two-matrix model. This model is an example of a larger class of models, "multi-matrix models", which aim to generalize the Hermitian one-matrix model. The interaction between matrices in multi-matrix models makes the analysis of the eigenvalue distributions more difficult than those for the one-matrix model. This dissertation makes use of a recent result which connects the interaction term for the two-matrix model to a combinatorial geometry problem. In particular, the interaction term is written as the Harish-Chandra-Itzykson-Zuber integral, which is expanded in a neighborhood of \tau=0, where \tau is the coupling constant appearing in the interaction term. The expansion involves monotone double Hurwitz numbers, which count a collection of ramified coverings of the two-sphere. Using this expansion, and its connection to monotone double Hurwitz numbers, this dissertation purports to describe a variational problem which describes the joint eigenvalue densities for the two-matrix model up to order \tau^2.