Percolação Bootstrap em árvores homogêneas

Abstract

Given a homogeneous tree with degree (...), and a initial density p of occupied sites, it is known that there exists a point (...) for wich the the final configuration of the bootstrap percolation model with threshold (...) in this tree shows two distinct phases for almost every initial configuration: it will have density of occupied vertices less than (...) and it will be entirely occupied if (...). In this work, besides of showing this result, we study another critical point related to this model. We show that there exists a point (...) which also divides all possible final configurations in two distinct cases for almost every initial configuration of this model: if (...), then we will have the occurrence of infinite clusters of occupied vertices and, if (...), then no infinite cluster can be found. In addition, we show that in the subcritical phase (...), the distribution of the occupied cluster size in the final bootstrapped configuration has an exponetial decay and show that, in this same final configuration, in the critial value (...) the expected occupied cluster size is infinite.