Estudo sobre a janela crítica de percolação em lages

Abstract

We study a independent Bernoulli percolation model in Zˆd through the work of H. Duminil-Copin, G. Kozma and V. Tassion on Upper bounds on the percolation correlation length (UB). The main result of this work is to quantify a known result of G. R. Grimmett and J. M. Marstrand in The Supercritical Phase of Percolation is Well Behaved (SPP). In UB is shown that there is an infinite cluster on Slab^d_n with parameter p_n+C\sqrt{\log n}, where p_n is related with the correlation length. Even more, the probability that the origin percolates on Slab^d_n for this parameter is greater than 1/2\sqrt{\log n}. This result is achieved using a different event than the one used in SPP and we discuss some differences, but with focus on the new work.