{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/97544"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/97544","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Applications of Stein's method and large deviations principle's in mean-field O(N) models","abstract":"In the first part of this thesis, we will discuss the classical XY model on complete graph in the mean-field (infinite-vertex) limit. Using theory of large deviations and Stein's method, in particular, Cramér and Sanov-type results, we present a number of results coming from the limit theorems with rates of convergence, and phase transition behavior for classical XY model. In the second part, we will generalize our results to mean-field classical $N$-vector models, for integers $N \\ge 2$. We will use the theory of large deviations and Stein's method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Some of the important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in [KM13] but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.","abstract_html":"In the first part of this thesis, we will discuss the classical XY model on complete graph in the mean-field (infinite-vertex) limit. Using theory of large deviations and Stein's method, in particular, Cramér and Sanov-type results, we present a number of results coming from the limit theorems with rates of convergence, and phase transition behavior for classical XY model. In the second part, we will generalize our results to mean-field classical $N$-vector models, for integers $N \\ge 2$. We will use the theory of large deviations and Stein's method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Some of the important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in [KM13] but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.","abstract_has_math":true,"creators":["Nawaz, Tayyab"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Kirkpatrick, Kay","DeVille, Lee","Rapti, Zoi","Hirani, Anil"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-08-10T19:51:44Z","date_published":"2017-08-10T19:51:44Z","updated_at":"2026-07-22T22:24:34Z","subjects":["Mean-field","Rate function","Total spin","Limit theorem","Phase transition"],"languages":["en"],"rights":["Copyright 2017 Tayyab Nawaz"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/97544","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kirkpatrick, Kay","DeVille, Lee","Rapti, Zoi","Hirani, Anil"]},{"key":"dc:creator","label":"Author","values":["Nawaz, Tayyab"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-08-10T19:51:44Z","2019-08-11T09:15:39Z","2017-04-07","2017-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mean-field","Rate function","Total spin","Limit theorem","Phase transition"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Tayyab Nawaz"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/97544"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In the first part of this thesis, we will discuss the classical XY model on complete graph in the mean-field (infinite-vertex) limit. Using theory of large deviations and Stein's method, in particular, Cramér and Sanov-type results, we present a number of results coming from the limit theorems with rates of convergence, and phase transition behavior for classical XY model. In the second part, we will generalize our results to mean-field classical $N$-vector models, for integers $N \\ge 2$. We will use the theory of large deviations and Stein's method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Some of the important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in [KM13] but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2019-05-01","The student, Tayyab Nawaz, accepted the attached license on 2017-03-30 at 17:49.","The student, Tayyab Nawaz, submitted this Dissertation for approval on 2017-03-30 at 17:50.","This Dissertation was approved for publication on 2017-04-07 at 11:08.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10626 on 2017-08-10 at 14:30:05","Made available in DSpace on 2017-08-10T19:51:44Z (GMT). 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Using theory of large deviations and Stein's method, in particular, Cramér and Sanov-type results, we present a number of results coming from the limit theorems with rates of convergence, and phase transition behavior for classical XY model. In the second part, we will generalize our results to mean-field classical $N$-vector models, for integers $N \\ge 2$. We will use the theory of large deviations and Stein's method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Some of the important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in [KM13] but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2019-05-01","The student, Tayyab Nawaz, accepted the attached license on 2017-03-30 at 17:49.","The student, Tayyab Nawaz, submitted this Dissertation for approval on 2017-03-30 at 17:50.","This Dissertation was approved for publication on 2017-04-07 at 11:08.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10626 on 2017-08-10 at 14:30:05","Made available in DSpace on 2017-08-10T19:51:44Z (GMT). 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