On the phase transition for some percolation models in random environments

• Ano de defesa: 2019
• Autor(a) principal: Marcos Vinícius Araújo Sá
• Orientador(a): Remy de Paiva Sanchis , Marcelo Richard Hilário
• Banca de defesa: Augusto Quadros Teixeira, Glauco Valle da Silva Coelho, Hubert Lacoin, Paulo Cupertino de Lima
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de Minas Gerais
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática
• Departamento: ICX - DEPARTAMENTO DE MATEMÁTICA
• País: Brasil
• Palavras-chave em Português: Percolation; Phase transition; Random environments; Renormalization; Multiscale analysis
• Link de acesso: http://hdl.handle.net/1843/50999

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 $\eta<1$.