{"id":{"repo_id":"arizona-thes","oai_identifier":"oai:repository.arizona.edu:10150/556704"},"canonical_url":"https://search.dev.ndltd.org/etd/arizona-thes/oai:repository.arizona.edu:10150/556704","repository":{"repo_id":"arizona-thes","name":"University of Arizona","base_url":"https://repository.arizona.edu/oai/request"},"display":{"title":"Combinatorics Of The Hermitian One-Matrix Model","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Waters, Patrick Thomas"],"institution":"The University of Arizona.","degree_name":"Ph.D.","degree_level":"doctoral","degree_discipline":"Graduate College","degree_department":null,"school":null,"contributors":[],"advisors":["Ercolani, Nicholas M."],"committee_chairs":[],"committee_members":["Ercolani, Nicholas M.","Kennedy, Tom G.","McLaughlin, Ken D.","Sethuraman, Sunder"],"year":2015,"date_issued":"2015","date_published":"2015","updated_at":"2026-07-24T00:56:22Z","subjects":["map","matrix","random","Mathematics","combinatorics"],"languages":["en_US"],"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."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10150/556704","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Ercolani, Nicholas M."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Ercolani, Nicholas M.","Kennedy, Tom G.","McLaughlin, Ken D.","Sethuraman, Sunder"]},{"key":"dc:creator","label":"Author","values":["Waters, Patrick Thomas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-06-10T21:57:19Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-06-10T21:57:19Z"]},{"key":"dc:date.issued","label":"Date","values":["2015"]},{"key":"dc:publisher","label":"Institution","values":["The University of Arizona."]},{"key":"dc:type","label":"Dc Type","values":["text","Electronic Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Graduate College","Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Arizona"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["map","matrix","random","Mathematics","combinatorics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10150/556704"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Combinatorics Of The Hermitian One-Matrix Model"]}]}],"canonical_facts":{"dc:contributor.advisor":["Ercolani, Nicholas M."],"dc:contributor.committeemember":["Ercolani, Nicholas M.","Kennedy, Tom G.","McLaughlin, Ken D.","Sethuraman, Sunder"],"dc:creator":["Waters, Patrick Thomas"],"dc:date.accessioned":["2015-06-10T21:57:19Z"],"dc:date.available":["2015-06-10T21:57:19Z"],"dc:date.issued":["2015"],"dc:description.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."],"dc:identifier.uri":["http://hdl.handle.net/10150/556704"],"dc:language.iso":["en_US"],"dc:publisher":["The University of Arizona."],"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."],"dc:subject":["map","matrix","random","Mathematics","combinatorics"],"dc:title":["Combinatorics Of The Hermitian One-Matrix Model"],"dc:type":["text","Electronic Dissertation"],"thesis:degree_discipline":["Graduate College","Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Arizona"]},"updated_at":"2026-07-24T00:56:22Z"}