{"id":{"repo_id":"iupui","oai_identifier":"oai:scholarworks.indianapolis.iu.edu:1805/45256"},"canonical_url":"https://search.dev.ndltd.org/etd/iupui/oai:scholarworks.indianapolis.iu.edu:1805/45256","repository":{"repo_id":"iupui","name":"IUPUI","base_url":"https://scholarworks.indianapolis.iu.edu/server/oai/request"},"display":{"title":"A Dynamical Approach to the Potts Model on Cayley Tree","abstract":"The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin σi ∈ {±1}. The q-state Potts model is a generalization of the Ising model, where each spin σi may take on q ≥3 number of states {0,··· ,q−1}. Both models have temperature T and an externally applied magnetic field h as parameters. Many statistical and physical properties of the q- state Potts model can be derived by studying its partition function. This includes phase transitions as T and/or h are varied. The celebrated Lee-Yang Theorem characterizes such phase transitions of the 2-state Potts model (the Ising model). This theorem does not hold for q > 2. Thus, phase transitions for the Potts model as h is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the 3-state Potts model as h is varied for constant T on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed T >0the 3-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of h or not at all, depending on T. However, an interesting new phenomenon occurs for the 3-state Potts model because the critical value of h can be non-zero for some range of temperatures. The 3-state Potts model for the antiferromagnetic case exhibits a phase transition at up to two critical values of h. The recursive constructions of the (n + 1)st level Cayley tree from two copies of the nth level Cayley tree allows one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems.","abstract_html":"The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin σi ∈ {±1}. The q-state Potts model is a generalization of the Ising model, where each spin σi may take on q ≥3 number of states {0,··· ,q−1}. Both models have temperature T and an externally applied magnetic field h as parameters. Many statistical and physical properties of the q- state Potts model can be derived by studying its partition function. This includes phase transitions as T and/or h are varied. The celebrated Lee-Yang Theorem characterizes such phase transitions of the 2-state Potts model (the Ising model). This theorem does not hold for q > 2. Thus, phase transitions for the Potts model as h is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the 3-state Potts model as h is varied for constant T on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed T >0the 3-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of h or not at all, depending on T. However, an interesting new phenomenon occurs for the 3-state Potts model because the critical value of h can be non-zero for some range of temperatures. The 3-state Potts model for the antiferromagnetic case exhibits a phase transition at up to two critical values of h. The recursive constructions of the (n + 1)st level Cayley tree from two copies of the nth level Cayley tree allows one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems.","abstract_has_math":false,"creators":["Pannipitiya, Diyath Nelaka"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Kitchens, Bruce P.","Roeder, Roland K. W."],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-12","date_published":"2024-12","updated_at":"2026-07-24T02:40:57Z","subjects":["Dynamical Systems","Renormalization Group Method","Potts Model","Lee-Yang Zeros","Ising Model"],"languages":["en_US"],"rights":["Attribution 4.0 International"],"rights_urls":["https://creativecommons.org/licenses/by/4.0"],"identifier_entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://doi.org/10.7912/NNFC-8W91"],"render_values":[{"text":"https://doi.org/10.7912/NNFC-8W91","href":"https://doi.org/10.7912/NNFC-8W91","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/1805/45256","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kitchens, Bruce P.","Roeder, Roland K. 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A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin σi ∈ {±1}. The q-state Potts model is a generalization of the Ising model, where each spin σi may take on q ≥3 number of states {0,··· ,q−1}. Both models have temperature T and an externally applied magnetic field h as parameters. Many statistical and physical properties of the q- state Potts model can be derived by studying its partition function. This includes phase transitions as T and/or h are varied. The celebrated Lee-Yang Theorem characterizes such phase transitions of the 2-state Potts model (the Ising model). This theorem does not hold for q > 2. Thus, phase transitions for the Potts model as h is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the 3-state Potts model as h is varied for constant T on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed T >0the 3-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of h or not at all, depending on T. However, an interesting new phenomenon occurs for the 3-state Potts model because the critical value of h can be non-zero for some range of temperatures. The 3-state Potts model for the antiferromagnetic case exhibits a phase transition at up to two critical values of h. The recursive constructions of the (n + 1)st level Cayley tree from two copies of the nth level Cayley tree allows one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems."]},{"key":"dc:title","label":"Title","values":["A Dynamical Approach to the Potts Model on Cayley Tree"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kitchens, Bruce P.","Roeder, Roland K. W."],"dc:contributor.other":["Geller, William","Perez, Rodrigo A."],"dc:creator":["Pannipitiya, Diyath Nelaka"],"dc:date.accessioned":["2025-01-13T09:28:13Z"],"dc:date.available":["2025-01-13T09:28:13Z"],"dc:date.issued":["2024-12"],"dc:description":["Indiana University-Purdue University Indianapolis (IUPUI)"],"dc:description.abstract":["The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin σi ∈ {±1}. The q-state Potts model is a generalization of the Ising model, where each spin σi may take on q ≥3 number of states {0,··· ,q−1}. Both models have temperature T and an externally applied magnetic field h as parameters. Many statistical and physical properties of the q- state Potts model can be derived by studying its partition function. This includes phase transitions as T and/or h are varied. The celebrated Lee-Yang Theorem characterizes such phase transitions of the 2-state Potts model (the Ising model). This theorem does not hold for q > 2. Thus, phase transitions for the Potts model as h is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the 3-state Potts model as h is varied for constant T on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed T >0the 3-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of h or not at all, depending on T. However, an interesting new phenomenon occurs for the 3-state Potts model because the critical value of h can be non-zero for some range of temperatures. The 3-state Potts model for the antiferromagnetic case exhibits a phase transition at up to two critical values of h. The recursive constructions of the (n + 1)st level Cayley tree from two copies of the nth level Cayley tree allows one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems."],"dc:identifier.uri":["https://hdl.handle.net/1805/45256","https://doi.org/10.7912/NNFC-8W91"],"dc:language.iso":["en_US"],"dc:rights":["Attribution 4.0 International"],"dc:rights.uri":["https://creativecommons.org/licenses/by/4.0"],"dc:subject":["Dynamical Systems","Renormalization Group Method","Potts Model","Lee-Yang Zeros","Ising Model"],"dc:title":["A Dynamical Approach to the Potts Model on Cayley Tree"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T02:40:57Z"}