Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range
Autor(a) principal: | |
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Data de Publicação: | 2024 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da USP |
Texto Completo: | https://doi.org/10.11606/T.45.2024.tde-07032024-191623 |
Resumo: | In this thesis, we present results on phase transition for two models: the semi-infinite Ising model with a decaying field, and the long-range Ising model with a random field. We study the semi-infinite Ising model with an external field h_i = λ |i_d|^{-δ}, λ is the wall influence, and δ>0. This external field decays as it gets further away from the wall. We are able to show that when δ>1 and \\β > \\β_c(d), there exists a critical value 0< λ_c:=λ_c(δ,\\β) such that, for λ<λ_c there is phase transition and for λ>λ_c we have uniqueness of the Gibbs state. In addition, when δ<1 we have only one Gibbs state for any positive \\β and λ. For the model with a random field, we extend the recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in the class of ferromagnetic random field Ising models. Our proof combines a generalization of Fröhlich-Spencer contours to the multidimensional setting proposed by Affonso, Bissacot, Endo and Handa, with the coarse-graining procedure introduced by Fisher, Fröhlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions J_=|x-y|^{- \\α} when \\α > d in dimension d>= 3 if we have a suitable system of contours, yielding an alternative proof that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen claimed that the RGM should also work on this generality. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions. |
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info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesis Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range Transição de fase em modelos de Ising: o semi-infinito com campo decaindo e o longo-alcance com campo aleatório 2024-02-07Rodrigo Bissacot ProençaLuiz Renato Goncalves FontesChristof KuelskeLeandro Chiarini MedeirosPierre PiccoJoão Vitor Teixeira MaiaUniversidade de São PauloMatemática AplicadaUSPBR Análise multiescala Campo aleatório Campo externo não-homogêneo Classical statistical mechanics Coarse-graining Coarse-graining Contornos Contours Energia livre de superfície Inhomogeneous external field Long-range random field Ising model Mecânica estatística clássica Modelo de Ising longo-alcance Modelo de Ising semi-infinito Multiscale analysis Phase transition Random field Semi-infinite Ising model surface free energy Transição de fase In this thesis, we present results on phase transition for two models: the semi-infinite Ising model with a decaying field, and the long-range Ising model with a random field. We study the semi-infinite Ising model with an external field h_i = λ |i_d|^{-δ}, λ is the wall influence, and δ>0. This external field decays as it gets further away from the wall. We are able to show that when δ>1 and \\β > \\β_c(d), there exists a critical value 0< λ_c:=λ_c(δ,\\β) such that, for λ<λ_c there is phase transition and for λ>λ_c we have uniqueness of the Gibbs state. In addition, when δ<1 we have only one Gibbs state for any positive \\β and λ. For the model with a random field, we extend the recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in the class of ferromagnetic random field Ising models. Our proof combines a generalization of Fröhlich-Spencer contours to the multidimensional setting proposed by Affonso, Bissacot, Endo and Handa, with the coarse-graining procedure introduced by Fisher, Fröhlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions J_=|x-y|^{- \\α} when \\α > d in dimension d>= 3 if we have a suitable system of contours, yielding an alternative proof that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen claimed that the RGM should also work on this generality. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions. Nesta tese apresentamos resultados referentes ao problema de transição de fase para dois modelos: o modelo de Ising semi-infinito com um campo decaindo e o modelo de Ising de longo-alcance com um campo aleatório. No modelo de Ising semi-infinito, o parâmetro relevante na existência de transição de fase é λ, a interação entre os spins do sistema e a parede que divide o lattice. Introduzindo um campo magnético que da forma h_i = λ |i_d|^{-δ} com δ>1, que decai conforme se afasta da parede, conseguimos mostrar que, em baixas temperaturas, o modelo ainda apresenta um ponto de criticalidade 0< λ_c(J,δ) satisfazendo: para 0<= λ <λ_c(J,δ) existem múltiplos estados de Gibbs e para λ>λ_c(J,δ) temos unicidade. Mostramos ainda que quando δ<1, λ_c(J, δ)=0 e portanto temos sempre unicidade. No modelo de Ising de longo-alcance com campo aleatório, estendemos um argumento de Ding e Zhuang do modelo de primeiros vizinhos para o modelo com interação de longo-alcance. Combinando uma generalização dos contornos de Fröhlich-Spencer, proposta por Affonso, Bissacot, Endo e Handa, com um procedimento de coarse graining introduzido por Fisher, Fröhlich, and Spencer, conseguimos provar que o modelo de Ising com interação J_=|x-y|^{- \\α} com \\α > d em dimensão d>= 3 apresenta transição de fase. Consideramos um campo aleatório dado por uma coleção i.i.d com distribuição Gaussiana ou Bernoulli. Nossa prova constitui uma prova alternativa que não usa grupos de renormalização (GR), uma vez que Bricmont e Kupiainen afirmaram que seus resultados usando GR funcionam para qualquer modelo que possua um sistema de contornos. https://doi.org/10.11606/T.45.2024.tde-07032024-191623info:eu-repo/semantics/openAccessengreponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USP2024-03-15T13:20:59Zoai:teses.usp.br:tde-07032024-191623Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212024-03-08T20:49:05Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false |
dc.title.en.fl_str_mv |
Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range |
dc.title.alternative.pt.fl_str_mv |
Transição de fase em modelos de Ising: o semi-infinito com campo decaindo e o longo-alcance com campo aleatório |
title |
Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range |
spellingShingle |
Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range João Vitor Teixeira Maia |
title_short |
Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range |
title_full |
Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range |
title_fullStr |
Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range |
title_full_unstemmed |
Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range |
title_sort |
Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range |
author |
João Vitor Teixeira Maia |
author_facet |
João Vitor Teixeira Maia |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Rodrigo Bissacot Proença |
dc.contributor.referee1.fl_str_mv |
Luiz Renato Goncalves Fontes |
dc.contributor.referee2.fl_str_mv |
Christof Kuelske |
dc.contributor.referee3.fl_str_mv |
Leandro Chiarini Medeiros |
dc.contributor.referee4.fl_str_mv |
Pierre Picco |
dc.contributor.author.fl_str_mv |
João Vitor Teixeira Maia |
contributor_str_mv |
Rodrigo Bissacot Proença Luiz Renato Goncalves Fontes Christof Kuelske Leandro Chiarini Medeiros Pierre Picco |
description |
In this thesis, we present results on phase transition for two models: the semi-infinite Ising model with a decaying field, and the long-range Ising model with a random field. We study the semi-infinite Ising model with an external field h_i = λ |i_d|^{-δ}, λ is the wall influence, and δ>0. This external field decays as it gets further away from the wall. We are able to show that when δ>1 and \\β > \\β_c(d), there exists a critical value 0< λ_c:=λ_c(δ,\\β) such that, for λ<λ_c there is phase transition and for λ>λ_c we have uniqueness of the Gibbs state. In addition, when δ<1 we have only one Gibbs state for any positive \\β and λ. For the model with a random field, we extend the recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in the class of ferromagnetic random field Ising models. Our proof combines a generalization of Fröhlich-Spencer contours to the multidimensional setting proposed by Affonso, Bissacot, Endo and Handa, with the coarse-graining procedure introduced by Fisher, Fröhlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions J_=|x-y|^{- \\α} when \\α > d in dimension d>= 3 if we have a suitable system of contours, yielding an alternative proof that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen claimed that the RGM should also work on this generality. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions. |
publishDate |
2024 |
dc.date.issued.fl_str_mv |
2024-02-07 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://doi.org/10.11606/T.45.2024.tde-07032024-191623 |
url |
https://doi.org/10.11606/T.45.2024.tde-07032024-191623 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade de São Paulo |
dc.publisher.program.fl_str_mv |
Matemática Aplicada |
dc.publisher.initials.fl_str_mv |
USP |
dc.publisher.country.fl_str_mv |
BR |
publisher.none.fl_str_mv |
Universidade de São Paulo |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da USP instname:Universidade de São Paulo (USP) instacron:USP |
instname_str |
Universidade de São Paulo (USP) |
instacron_str |
USP |
institution |
USP |
reponame_str |
Biblioteca Digital de Teses e Dissertações da USP |
collection |
Biblioteca Digital de Teses e Dissertações da USP |
repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP) |
repository.mail.fl_str_mv |
virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br |
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1794502343897645056 |