Persistent Link:
http://hdl.handle.net/10150/556704
Title:
Combinatorics Of The Hermitian One-Matrix Model
Author:
Waters, Patrick Thomas
Issue Date:
2015
Publisher:
The University of Arizona.
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.
Abstract:
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.
Type:
text; Electronic Dissertation
Keywords:
map; matrix; random; Mathematics; combinatorics
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Ercolani, Nicholas M.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleCombinatorics Of The Hermitian One-Matrix Modelen_US
dc.creatorWaters, Patrick Thomasen
dc.contributor.authorWaters, Patrick Thomasen
dc.date.issued2015en
dc.publisherThe University of Arizona.en
dc.rightsCopyright © 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.en
dc.description.abstractIt 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.en
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectmapen
dc.subjectmatrixen
dc.subjectrandomen
dc.subjectMathematicsen
dc.subjectcombinatoricsen
thesis.degree.namePh.D.en
thesis.degree.leveldoctoralen
thesis.degree.disciplineGraduate Collegeen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorUniversity of Arizonaen
dc.contributor.advisorErcolani, Nicholas M.en
dc.contributor.committeememberErcolani, Nicholas M.en
dc.contributor.committeememberKennedy, Tom G.en
dc.contributor.committeememberMcLaughlin, Ken D.en
dc.contributor.committeememberSethuraman, Sunderen
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