Enumeration of Ribbon Graphs Using Complex Gaussian Random Variables
Virgil Pierce, University of Texas Rio Grande Valley
Our work centers on the partition function of N-by-N Hermitian random matrices. This function encodes the expected values that one would like to compute in understanding the distribution of eigenvalues of these matrices as a probability distribution function. The function has a natural interpretation in terms of solutions of the Toda lattice hierarchy, a classical example of an integrable system, a system of ODEs that possesses a complete family of constants of motion. The asymptotic expansion of the log-partition function gives generating functions for enumeration of maps indexed by genus, and the parameters of the deformation index the maps by the valency of their vertices. In a broad class of cases we have been able to exploit the connection with the Toda lattice hierarchy to derive explicit expressions for the terms of the asymptotic expansion of the log-partition function. We will give an overview of the known results for map enumeration and their connection with a class of combinatoric problems related to generalizations of the Catalan Numbers.