{"id":{"repo_id":"iupui","oai_identifier":"oai:scholarworks.indianapolis.iu.edu:1805/22848"},"canonical_url":"https://search.dev.ndltd.org/etd/iupui/oai:scholarworks.indianapolis.iu.edu:1805/22848","repository":{"repo_id":"iupui","name":"IUPUI","base_url":"https://scholarworks.indianapolis.iu.edu/server/oai/request"},"display":{"title":"Some Connections Between Complex Dynamics and Statistical Mechanics","abstract":"Associated to any finite simple graph $\\Gamma$ is the {\\em chromatic polynomial} $\\P_\\Gamma(q)$ whose complex zeros are called the {\\em chromatic zeros} of $\\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\\{\\Gamma_n\\}_{n=0}^\\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\\mu_n$ converges to some measure $\\mu$ as $n$ tends to infinity. We call $\\mu$ the {\\em limiting measure of chromatic zeros} associated to $\\{\\Gamma_n\\}_{n=0}^\\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.","abstract_html":"Associated to any finite simple graph $\\Gamma$ is the {\\em chromatic polynomial} <span class=\"etd-inline-math\">\\P<sub>\\</sub>Gamma(q)</span> whose complex zeros are called the {\\em chromatic zeros} of $\\Gamma$. A hierarchical lattice is a sequence of finite simple graphs <span class=\"etd-inline-math\">\\{\\Gamma<sub>n</sub>\\}<sub>n=0</sub><sup>\\</sup>infty</span> built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let <span class=\"etd-inline-math\">&mu;<sub>n</sub></span> denote the probability measure that assigns a Dirac measure to each chromatic zero of <span class=\"etd-inline-math\">\\Gamma<sub>n</sub></span>. Under a mild hypothesis on the generating graph, we prove that the sequence <span class=\"etd-inline-math\">&mu;<sub>n</sub></span> converges to some measure <span class=\"etd-inline-math\">&mu;</span> as $n$ tends to infinity. We call <span class=\"etd-inline-math\">&mu;</span> the {\\em limiting measure of chromatic zeros} associated to <span class=\"etd-inline-math\">\\{\\Gamma<sub>n</sub>\\}<sub>n=0</sub><sup>\\</sup>infty</span>. In the case of the Diamond Hierarchical Lattice we prove that the support of <span class=\"etd-inline-math\">&mu;</span> has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.","abstract_has_math":true,"creators":["Chio, Ivan"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Roeder, Roland K. W."],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-05","date_published":"2020-05","updated_at":"2026-07-24T02:41:08Z","subjects":["Complex Dynamics","Dynamical Systems","Statistical Mechanics","Hierarchical Lattices"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://dx.doi.org/10.7912/C2/2412"],"render_values":[{"text":"http://dx.doi.org/10.7912/C2/2412","href":"http://dx.doi.org/10.7912/C2/2412","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/1805/22848","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Roeder, Roland K. W."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Misiurewicz, Michal","Perez, Rodrigo A.","Yattselev, Maxim L."]},{"key":"dc:creator","label":"Author","values":["Chio, Ivan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-05-21T17:14:59Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-05-21T17:14:59Z"]},{"key":"dc:date.issued","label":"Date","values":["2020-05"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Complex Dynamics","Dynamical Systems","Statistical Mechanics","Hierarchical Lattices"]}]},{"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/1805/22848","http://dx.doi.org/10.7912/C2/2412"]}]},{"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":["Associated to any finite simple graph $\\Gamma$ is the {\\em chromatic polynomial} $\\P_\\Gamma(q)$ whose complex zeros are called the {\\em chromatic zeros} of $\\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\\{\\Gamma_n\\}_{n=0}^\\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\\mu_n$ converges to some measure $\\mu$ as $n$ tends to infinity. We call $\\mu$ the {\\em limiting measure of chromatic zeros} associated to $\\{\\Gamma_n\\}_{n=0}^\\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree."]},{"key":"dc:title","label":"Title","values":["Some Connections Between Complex Dynamics and Statistical Mechanics"]}]}],"canonical_facts":{"dc:contributor.advisor":["Roeder, Roland K. W."],"dc:contributor.other":["Misiurewicz, Michal","Perez, Rodrigo A.","Yattselev, Maxim L."],"dc:creator":["Chio, Ivan"],"dc:date.accessioned":["2020-05-21T17:14:59Z"],"dc:date.available":["2020-05-21T17:14:59Z"],"dc:date.issued":["2020-05"],"dc:description":["Indiana University-Purdue University Indianapolis (IUPUI)"],"dc:description.abstract":["Associated to any finite simple graph $\\Gamma$ is the {\\em chromatic polynomial} $\\P_\\Gamma(q)$ whose complex zeros are called the {\\em chromatic zeros} of $\\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\\{\\Gamma_n\\}_{n=0}^\\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\\mu_n$ converges to some measure $\\mu$ as $n$ tends to infinity. We call $\\mu$ the {\\em limiting measure of chromatic zeros} associated to $\\{\\Gamma_n\\}_{n=0}^\\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree."],"dc:identifier.uri":["https://hdl.handle.net/1805/22848","http://dx.doi.org/10.7912/C2/2412"],"dc:language.iso":["en_US"],"dc:subject":["Complex Dynamics","Dynamical Systems","Statistical Mechanics","Hierarchical Lattices"],"dc:title":["Some Connections Between Complex Dynamics and Statistical Mechanics"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T02:41:08Z"}