Explosive transitions in site percolation

Due to its simplicity and great application, it is known that percolation is one of the phenomena widely studied by statistical physics that addresses the theory of phase transitions and critical phenomena. In 2009 [1] the authors proposed a percolation variant introducing a competitive process between sites (or bonds), which prevents large clusters from joining each other, as a possible means of delaying the phase transition of a densely connected network, leading to explosive transitions or atypical and abnormal behaviors. This new type of percolation brought great interest, leading to a series of studies, among which the analysis of the order of the transition (continuous or discontinuous), the creation of others models with explosive behaviors, scale analysis, among others.
In this Master thesis we will study site percolation. As an algorithm to delay the transition and cause explosive percolation, we propose a variant of the sum rule proposed in [D. Achlioptas, R.M. D’Souza, J. Spencer, Science, 323,1453, (2009)] which we call global sum rule. In order to characterize the phase transition we will make use of numerical analysis. We explore the behavior of the transition for different order parameters, in the same way we evaluate the changes that the transition can undergo with different sizes and dimensions of the network, as well as for different number of tries in the global sum rule.