Probability of incipient spanning clusters in critical square bond percolation

摘要

The probability of simultaneous occurrence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice with free boundaries at the bond percolation threshold p(c) = 1/2. It is found that the probability of k and more Incipient Spanning Clusters (ISC) have the values P(k > 1) approximate to 0.00658(3) and P(k > 2) approximate to 0.00000148(21) provided that the limit of these probabilities for infinite lattices exists. The probability P(k > 3) of more than three ISC could be estimated to be of the order of 10(-11) and is beyond the possibility to compute such a value by nowadays computers. So, it is impossible to check in simulations the Aizenman law for the probabilities when k much greater than 1. We have detected a single sample with four ISC in a total number of about 10(10) samples investigated. The probability of this single event is 1/10 for that number of samples. The influence of boundary conditions is discussed in the last section.