Combinatorics Of The Hermitian One-Matrix Model

It is well known that the perturbed GUE matrix model has a combinatorial interpretation involving graphs embedded in Riemann surfaces. Generating functions for these graphs in the case of an even potential have been studied by many authors. The case of a cubic potential has also been studied. Using string equations, we construct "valence independent" formulas for map generating functions. These formulas hold for arbitrary polynomial potentials. We derive "edge Toda equations," which we use together with our valence independent formulas to generalize formulas of Ercolani, McLaughlin and Pierce to the case of an arbitrary odd or even valence. We derive a valence independent formula for the equilibrium measure for eigenvalues of the matrix model. Using this formula for the equilibrium measure we show that our valence independent formulas for generating functions can also be derived from the Riemann-Hilbert problem for orthogonal polynomials, and from the loop equations.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Graduate College

Grantor dc:publisher

The University of Arizona.

Year dc:date.issued

2015

Author dc:creator

• Waters, Patrick Thomas

Advisor dc:contributor.advisor

• Ercolani, Nicholas M.

Committee members dc:contributor.committeemember

• Ercolani, Nicholas M.
• Kennedy, Tom G.
• McLaughlin, Ken D.
• Sethuraman, Sunder

• map
• matrix
• random
• Mathematics
• combinatorics

Statement dc:rights

• Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/10150/556704

OAI identifier oai:identifier

oai:repository.arizona.edu:10150/556704