Probability of incipient spanning clusters in critical square bond percolation

Journal Article · Sun Jun 01 04:00:00 EDT 1997 · International Journal of Modern Physics C

DOI:https://doi.org/10.1142/S0129183197000394· OSTI ID:535104

Shchur, L N; Kosyakov, S S ^[1]

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1. Landau Institute for Theoretical Physics, Chernogolovka (Russian Federation)

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{sub c} = {1/2}. It is found that the probability of k and more Incipient Spanning Clusters (ISC) have the values P(k > 1) {approx} 0.00658(3) and P(k > 2) {approx} 0.00000148(21) provided that the limit of these probabilities for infinite lattice exists. The probability P(k > 3) of more than three ISC could be estimated to be of the order of 10{sup -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_gt} 1. We have detected a single sample with four ISC in a total number of about 10{sup 10} samples investigated. The probability of this single event is 1/10 for the number of samples. The influence of boundary conditions is discussed in the last section.

• OSTI ID: 535104
• Journal Information: International Journal of Modern Physics C, Journal Name: International Journal of Modern Physics C Journal Issue: 3 Vol. 8; ISSN 0129-1831; ISSN IJMPEO
• Country of Publication: United States
• Language: English

Related Subjects

40 CHEMISTRY
66 PHYSICS
99 GENERAL AND MISCELLANEOUS
ALGORITHMS
BOUNDARY CONDITIONS
CHEMICAL BONDS
MONTE CARLO METHOD
PROBABILITY
SOLID CLUSTERS