Probability of Incipient Spanning Clusters in Critical Square Bond Percolation
Abstract
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) ≈ 0.00658(3) and P(k>2) ≈ 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≫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.
 Publication:

International Journal of Modern Physics C
 Pub Date:
 1997
 DOI:
 10.1142/S0129183197000394
 arXiv:
 arXiv:condmat/9702248
 Bibcode:
 1997IJMPC...8..473S
 Keywords:

 Monte Carlo Simulation;
 Coexistence of Spanning Clusters;
 Aizenman Theory;
 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Lattice
 EPrint:
 7 pages, 1 table, 5 figures (1PS+4*Latex),uses epsf.sty Int.J.Mod.Phys. C (submitted to)