Modelo de percolação bidimensional com dependência local

• Autor(a) principal: Oliveira, Jaelson dos Santos
• Data de Publicação: 2018
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações da UFPB
• Texto Completo: https://repositorio.ufpb.br/jspui/handle/123456789/13332
• topic: Percolação; Grafos; Grafos planares; CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO

In our work we will present a brief historical summary of the emergence of the abstract mathematical object called graph. In addition to concepts, definitions and their theories with day-to-day applications and models used either intuitively or not. We will know the definition of planar graphs. We can highlight the case of solving an electrical system using graphs or social networks. Deepening our studies, we will enter into the subject of percolation. The original percolation model involves the points Z2 where each point can have an open or closed edge connected to each of its neighbors with probability p independently of each other. Our study will also take place in the two - dimensional plane or in the Z2. We say that there is percolation when there is a positive probability that a random path will occur in the process from the origin with an infinite number of edges. Our probabilistic model consists of two types of fluids (blue and red colors). Different from the initial model, we traverse the vertices of Z2 in a deterministic manner (spiral counterclockwise). At each vertex we draw a direction that has not yet been occupied, chosen uniformly, then we draw the size of the link that will be constructed, Suchalinkcanberepresentedbyanedgeofsize 0.1 and 2, respectively, to the link (size 0), or a red edge of size 1 or size 2 blue. The size of each link is given by a random binomial variable of parameters n = 2 and p. It is worth mentioning that in our model there is a high dependence between neighboring links. Our intuition is that there are two critical values for the model, we call them pv and pa, where for every value of p < pv there is no percolation, for pv < p < pa there is percolation for both red and blue fluid, and in the case of p > pa there is only percolation to the blue fluid. In order to better visualize the physical process, we simulate the model at different scales and for different values of p and using a graphical construct.