PROBABILITY OF INCIPIENT SPANNING CLUSTERS IN CRITICAL SQUARE BOND PERCOLATION

The probability of simultaneous occurrence of at least k spanning clus ters has been studied by Monte Carlo simulations on the 2D square latt ice with free boundaries at the bond percolation threshold p(c) = 1/2. It is found that the probability of k and more Incipient Spanning Clu sters (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 p robabilities 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 comp uters. So, it is impossible to check in simulations the Aizenman law f or the probabilities when k much greater than 1. We have detected a si ngle sample with four ISC in a total number of about 10(10) samples in vestigated. The probability of this single event is 1/10 for that numb er of samples. The influence of boundary conditions is discussed in th e last section.