On the phase transition for some percolation models in random environments
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFMG |
Texto Completo: | http://hdl.handle.net/1843/50999 |
Resumo: | In this thesis we consider two percolation models in random environments and we are interested in their phase transition phenomenon. The first percolation model we study is defined on the cubic lattice featuring columnar disorder. This model is defined in two steps: first the vertical columns of $\mathbb{Z}^3$ are removed independently with probability $1-\rho$ and, in the second step, the bonds connecting sites in the remaining sub-lattice are declared open with probability $p$, independently. Our result shows that there exists $\delta>0$ such that $p_c(\rho)<1/2-\delta$ for any $\rho>\rho_c$, where $\rho_c$ denotes the critical point of site percolation in $\mathbb{Z}^2$. The second model is defined on a horizontally stretched square lattice, which is a generalized version of $\mathbb{Z}^2_+$ obtained by stretching the distances between its columns according to a positive random variable $\xi$. In this model the probability of a bond being declared open will decay exponentially according to its length. Our result shows the existence of a phase transition when $\mathbb{E}(\xi^\eta)<\infty$, for some $\eta>1$, and the absence of phase transition when $\mathbb{E}(\xi^\eta)=\infty$ for some $\eta<1$. |
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Remy de Paiva Sanchishttp://lattes.cnpq.br/1582551703060830Marcelo Richard HilárioAugusto Quadros TeixeiraGlauco Valle da Silva CoelhoHubert LacoinPaulo Cupertino de Limahttp://lattes.cnpq.br/0158987925235945Marcos Vinícius Araújo Sá2023-03-17T16:33:11Z2023-03-17T16:33:11Z2019-11-22http://hdl.handle.net/1843/50999In this thesis we consider two percolation models in random environments and we are interested in their phase transition phenomenon. The first percolation model we study is defined on the cubic lattice featuring columnar disorder. This model is defined in two steps: first the vertical columns of $\mathbb{Z}^3$ are removed independently with probability $1-\rho$ and, in the second step, the bonds connecting sites in the remaining sub-lattice are declared open with probability $p$, independently. Our result shows that there exists $\delta>0$ such that $p_c(\rho)<1/2-\delta$ for any $\rho>\rho_c$, where $\rho_c$ denotes the critical point of site percolation in $\mathbb{Z}^2$. The second model is defined on a horizontally stretched square lattice, which is a generalized version of $\mathbb{Z}^2_+$ obtained by stretching the distances between its columns according to a positive random variable $\xi$. In this model the probability of a bond being declared open will decay exponentially according to its length. Our result shows the existence of a phase transition when $\mathbb{E}(\xi^\eta)<\infty$, for some $\eta>1$, and the absence of phase transition when $\mathbb{E}(\xi^\eta)=\infty$ for some $\eta<1$.Nesta tese nós consideramos dois modelos de percolação em ambientes aleatórios e estamos interessados em seus fenômenos de transição de fase. O primeiro modelo de percolação estudado é na rede cúbica apresentando desordem colunar. Este modelo é definido em dois passos: primeiro as colunas verticais de $\mathbb{Z}^3$ são removidas independentemente com probabilidade $1-\rho$ e, no segundo passo, os elos conectando sítios na sub-rede remanescente são declarados abertos com probabilidade $p$ de modo independente. Nosso resultado mostra que existe $\delta>0$ tal que o ponto crítico $p_c(\rho)<1/2-\delta$ para todo $\rho>\rho_c$, onde $\rho_c$ denota o ponto crítico da percolação de sítios em $\mathbb{Z}^2$. O segundo modelo é na rede quadrada esticada horizontalmente, que consiste de uma versão generalizada de $\mathbb{Z}^2_+$ obtida ao se esticar a distância entre suas colunas, segundo uma variável aleatória positiva $\xi$. Neste modelo a probabilidade de um elo ser declarado aberto decairá exponencialmente segundo seu comprimento. Nosso resultado mostra a existência da transição de fase quando $\mathbb{E}(\xi^\eta)<\infty$, para algum $\eta>1$, e a ausência quando $\mathbb{E}(\xi^\eta)=\infty$, para algum $\eta<1$.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorengUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICAMatemática – TesesPercolação – TesesTransição de fase – TesesCampos aleatórios – TesesGrupo de renormalização– TesesPercolationPhase transitionRandom environmentsRenormalizationMultiscale analysisOn the phase transition for some percolation models in random environmentsA transição de fase para alguns modelos de percolação em ambientes aleatóriosinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALOn_the_phase_transition_for_some_percolation_models_in_random_environments.pdfOn_the_phase_transition_for_some_percolation_models_in_random_environments.pdfapplication/pdf889861https://repositorio.ufmg.br/bitstream/1843/50999/1/On_the_phase_transition_for_some_percolation_models_in_random_environments.pdf7802e4ae9c7f3ee83fa65be75d5407a4MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-82118https://repositorio.ufmg.br/bitstream/1843/50999/2/license.txtcda590c95a0b51b4d15f60c9642ca272MD521843/509992023-03-17 13:33:12.2oai:repositorio.ufmg.br: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ório de PublicaçõesPUBhttps://repositorio.ufmg.br/oaiopendoar:2023-03-17T16:33:12Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false |
dc.title.pt_BR.fl_str_mv |
On the phase transition for some percolation models in random environments |
dc.title.alternative.pt_BR.fl_str_mv |
A transição de fase para alguns modelos de percolação em ambientes aleatórios |
title |
On the phase transition for some percolation models in random environments |
spellingShingle |
On the phase transition for some percolation models in random environments Marcos Vinícius Araújo Sá Percolation Phase transition Random environments Renormalization Multiscale analysis Matemática – Teses Percolação – Teses Transição de fase – Teses Campos aleatórios – Teses Grupo de renormalização– Teses |
title_short |
On the phase transition for some percolation models in random environments |
title_full |
On the phase transition for some percolation models in random environments |
title_fullStr |
On the phase transition for some percolation models in random environments |
title_full_unstemmed |
On the phase transition for some percolation models in random environments |
title_sort |
On the phase transition for some percolation models in random environments |
author |
Marcos Vinícius Araújo Sá |
author_facet |
Marcos Vinícius Araújo Sá |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Remy de Paiva Sanchis |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/1582551703060830 |
dc.contributor.advisor2.fl_str_mv |
Marcelo Richard Hilário |
dc.contributor.referee1.fl_str_mv |
Augusto Quadros Teixeira |
dc.contributor.referee2.fl_str_mv |
Glauco Valle da Silva Coelho |
dc.contributor.referee3.fl_str_mv |
Hubert Lacoin |
dc.contributor.referee4.fl_str_mv |
Paulo Cupertino de Lima |
dc.contributor.authorLattes.fl_str_mv |
http://lattes.cnpq.br/0158987925235945 |
dc.contributor.author.fl_str_mv |
Marcos Vinícius Araújo Sá |
contributor_str_mv |
Remy de Paiva Sanchis Marcelo Richard Hilário Augusto Quadros Teixeira Glauco Valle da Silva Coelho Hubert Lacoin Paulo Cupertino de Lima |
dc.subject.por.fl_str_mv |
Percolation Phase transition Random environments Renormalization Multiscale analysis |
topic |
Percolation Phase transition Random environments Renormalization Multiscale analysis Matemática – Teses Percolação – Teses Transição de fase – Teses Campos aleatórios – Teses Grupo de renormalização– Teses |
dc.subject.other.pt_BR.fl_str_mv |
Matemática – Teses Percolação – Teses Transição de fase – Teses Campos aleatórios – Teses Grupo de renormalização– Teses |
description |
In this thesis we consider two percolation models in random environments and we are interested in their phase transition phenomenon. The first percolation model we study is defined on the cubic lattice featuring columnar disorder. This model is defined in two steps: first the vertical columns of $\mathbb{Z}^3$ are removed independently with probability $1-\rho$ and, in the second step, the bonds connecting sites in the remaining sub-lattice are declared open with probability $p$, independently. Our result shows that there exists $\delta>0$ such that $p_c(\rho)<1/2-\delta$ for any $\rho>\rho_c$, where $\rho_c$ denotes the critical point of site percolation in $\mathbb{Z}^2$. The second model is defined on a horizontally stretched square lattice, which is a generalized version of $\mathbb{Z}^2_+$ obtained by stretching the distances between its columns according to a positive random variable $\xi$. In this model the probability of a bond being declared open will decay exponentially according to its length. Our result shows the existence of a phase transition when $\mathbb{E}(\xi^\eta)<\infty$, for some $\eta>1$, and the absence of phase transition when $\mathbb{E}(\xi^\eta)=\infty$ for some $\eta<1$. |
publishDate |
2019 |
dc.date.issued.fl_str_mv |
2019-11-22 |
dc.date.accessioned.fl_str_mv |
2023-03-17T16:33:11Z |
dc.date.available.fl_str_mv |
2023-03-17T16:33:11Z |
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 |
http://hdl.handle.net/1843/50999 |
url |
http://hdl.handle.net/1843/50999 |
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 Federal de Minas Gerais |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Matemática |
dc.publisher.initials.fl_str_mv |
UFMG |
dc.publisher.country.fl_str_mv |
Brasil |
dc.publisher.department.fl_str_mv |
ICX - DEPARTAMENTO DE MATEMÁTICA |
publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFMG instname:Universidade Federal de Minas Gerais (UFMG) instacron:UFMG |
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UFMG |
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Repositório Institucional da UFMG |
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Repositório Institucional da UFMG |
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