Transição de fase no modelo de percolação independente em grafos

Abstract

In this text will be studied the phase transition of the independent percolation model in graphs. In the first chapter will be detailed Lyons' work [11], in which is related the process of percolation with random walks and electrical circuits in order to prove that the critical point of a tree is equal to the inverse of its branching number. The second chapteris intended to the percolation study in more general graphs. In its first section, we show that the graphs with limited degree percolades at the vertices if and only if, percolades at the edges, and also prove that graphs with positive Cheeger constant percolades. In the other sections we will quote the value of the critical point through Peierls argument, showing a large family of graphs that percolades. We will be based on the following articles: Alves, Procacci e Sanchis [1], Kozma [9] and Timár [13].