{"id":{"repo_id":"iupui","oai_identifier":"oai:scholarworks.indianapolis.iu.edu:1805/2482"},"canonical_url":"https://search.dev.ndltd.org/etd/iupui/oai:scholarworks.indianapolis.iu.edu:1805/2482","repository":{"repo_id":"iupui","name":"IUPUI","base_url":"https://scholarworks.indianapolis.iu.edu/server/oai/request"},"display":{"title":"Exact Solutions to the Six-Vertex Model with Domain Wall Boundary Conditions and Uniform Asymptotics of Discrete Orthogonal Polynomials on an Infinite Lattice","abstract":"In this dissertation the partition function, $Z_n$, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that $Z_n=CG^nF^{n^2}(1+O(e^{-n^{1-\\ep}}))$ for any $\\ep>0$, and we give explicit formulae for the numbers $C, G$, and $F$. On the critical line separating the ferroelectric and disordered phase regions, we show that $Z_n=Cn^{1/4}G^{\\sqrt{n}}F^{n^2}(1+O(n^{-1/2}))$, and we give explicit formulae for the numbers $G$ and $F$. In this phase region, the value of the constant $C$ is unknown. In the antiferroelectric phase region, we show that $Z_n=C\\th_4(n\\om)F^{n^2}(1+O(n^{-1}))$, where $\\th_4$ is Jacobi's theta function, and explicit formulae are given for the numbers $\\om$ and $F$. The value of the constant $C$ is unknown in this phase region. In each case, the proof is based on reformulating $Z_n$ as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large $n$ asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation.","abstract_html":"In this dissertation the partition function, <span class=\"etd-inline-math\">Z<sub>n</sub></span>, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that <span class=\"etd-inline-math\">Z<sub>n</sub>=CG<sup>n</sup>F<sup>n<sup>2</sup></sup>(1+O(e<sup>-n<sup>1-\\ep</sup></sup>))</span> for any $\\ep&gt;0$, and we give explicit formulae for the numbers $C, G$, and $F$. On the critical line separating the ferroelectric and disordered phase regions, we show that <span class=\"etd-inline-math\">Z<sub>n</sub>=Cn<sup>1/4</sup>G<sup>\\sqrt{n}</sup>F<sup>n<sup>2</sup></sup>(1+O(n<sup>-1/2</sup>))</span>, and we give explicit formulae for the numbers $G$ and $F$. In this phase region, the value of the constant $C$ is unknown. In the antiferroelectric phase region, we show that <span class=\"etd-inline-math\">Z<sub>n</sub>=C\\th<sub>4</sub>(n\\om)F<sup>n<sup>2</sup></sup>(1+O(n<sup>-1</sup>))</span>, where <span class=\"etd-inline-math\">\\th<sub>4</sub></span> is Jacobi&#x27;s theta function, and explicit formulae are given for the numbers $\\om$ and $F$. The value of the constant $C$ is unknown in this phase region. In each case, the proof is based on reformulating <span class=\"etd-inline-math\">Z<sub>n</sub></span> as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large $n$ asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation.","abstract_has_math":true,"creators":["Liechty, Karl Edmund"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Bleher, Pavel, 1947-"],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-03-09","date_published":"2011-03-09","updated_at":"2026-07-24T02:41:52Z","subjects":["Statistical Mechanics, Random Matrices, Orthogonal Polynomials, Asymptotics, Riemann-Hilbert Problems"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://dx.doi.org/10.7912/C2/2390"],"render_values":[{"text":"http://dx.doi.org/10.7912/C2/2390","href":"http://dx.doi.org/10.7912/C2/2390","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/1805/2482","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Bleher, Pavel, 1947-"]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Its, Alexander R.","Lempert, Lazlo","Kitchens, Bruce, 1953-"]},{"key":"dc:creator","label":"Author","values":["Liechty, Karl Edmund"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2011-03-09T19:38:44Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2011-03-09T19:38:44Z"]},{"key":"dc:date.issued","label":"Date","values":["2011-03-09"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Statistical Mechanics, Random Matrices, Orthogonal Polynomials, Asymptotics, Riemann-Hilbert Problems"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1805/2482","http://dx.doi.org/10.7912/C2/2390"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Indiana University-Purdue University Indianapolis (IUPUI)"]},{"key":"dc:description.abstract","label":"Abstract","values":["In this dissertation the partition function, $Z_n$, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that $Z_n=CG^nF^{n^2}(1+O(e^{-n^{1-\\ep}}))$ for any $\\ep>0$, and we give explicit formulae for the numbers $C, G$, and $F$. On the critical line separating the ferroelectric and disordered phase regions, we show that $Z_n=Cn^{1/4}G^{\\sqrt{n}}F^{n^2}(1+O(n^{-1/2}))$, and we give explicit formulae for the numbers $G$ and $F$. In this phase region, the value of the constant $C$ is unknown. In the antiferroelectric phase region, we show that $Z_n=C\\th_4(n\\om)F^{n^2}(1+O(n^{-1}))$, where $\\th_4$ is Jacobi's theta function, and explicit formulae are given for the numbers $\\om$ and $F$. The value of the constant $C$ is unknown in this phase region. In each case, the proof is based on reformulating $Z_n$ as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large $n$ asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation."]},{"key":"dc:title","label":"Title","values":["Exact Solutions to the Six-Vertex Model with Domain Wall Boundary Conditions and Uniform Asymptotics of Discrete Orthogonal Polynomials on an Infinite Lattice"]}]}],"canonical_facts":{"dc:contributor.advisor":["Bleher, Pavel, 1947-"],"dc:contributor.other":["Its, Alexander R.","Lempert, Lazlo","Kitchens, Bruce, 1953-"],"dc:creator":["Liechty, Karl Edmund"],"dc:date.accessioned":["2011-03-09T19:38:44Z"],"dc:date.available":["2011-03-09T19:38:44Z"],"dc:date.issued":["2011-03-09"],"dc:description":["Indiana University-Purdue University Indianapolis (IUPUI)"],"dc:description.abstract":["In this dissertation the partition function, $Z_n$, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that $Z_n=CG^nF^{n^2}(1+O(e^{-n^{1-\\ep}}))$ for any $\\ep>0$, and we give explicit formulae for the numbers $C, G$, and $F$. On the critical line separating the ferroelectric and disordered phase regions, we show that $Z_n=Cn^{1/4}G^{\\sqrt{n}}F^{n^2}(1+O(n^{-1/2}))$, and we give explicit formulae for the numbers $G$ and $F$. In this phase region, the value of the constant $C$ is unknown. In the antiferroelectric phase region, we show that $Z_n=C\\th_4(n\\om)F^{n^2}(1+O(n^{-1}))$, where $\\th_4$ is Jacobi's theta function, and explicit formulae are given for the numbers $\\om$ and $F$. The value of the constant $C$ is unknown in this phase region. In each case, the proof is based on reformulating $Z_n$ as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large $n$ asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation."],"dc:identifier.uri":["https://hdl.handle.net/1805/2482","http://dx.doi.org/10.7912/C2/2390"],"dc:language.iso":["en_US"],"dc:subject":["Statistical Mechanics, Random Matrices, Orthogonal Polynomials, Asymptotics, Riemann-Hilbert Problems"],"dc:title":["Exact Solutions to the Six-Vertex Model with Domain Wall Boundary Conditions and Uniform Asymptotics of Discrete Orthogonal Polynomials on an Infinite Lattice"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T02:41:52Z"}