Percolação de longo alcance em grafos hierárquicos

Abstract

In this dissertation the percolation model of the article Long-range percolation on the hierarchical lattice [18] by Koval, Meester and Trapman will be presented in detail. In the long-range percolation model, any two vertices may be connected by an edge with probability 1 exp{k}, where and are parameters of the model and k is the distance between the vertices. This distance will depend on an integer parameter N 2 which defines the hierarchy on the model. Given a configuration W of edges that are open or not on the lattice, it is possible to study the open cluster of the origin C(0; w), i.e., the set of vertices that are connected to the origin by an open path. The probability that the open cluster of the origin is infinite is denoted by (,). The main results of the paper [18] and detailed explanations of proofs of theorems are in Chapters 2 to 6. Among them are the phase transition of the model, the uniqueness of the infinite cluster and continuity of functions (,) and c() := inf{; (,) > 0}.