Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

Lev N. Shchur () and Sergey S. Kosyakov
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Lev N. Shchur: Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia
Sergey S. Kosyakov: Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

International Journal of Modern Physics C (IJMPC), 1997, vol. 08, issue 03, 473-481

Abstract: The probability of simultaneous occurrence of at leastkspanning clusters has been studied by Monte Carlo simulations on the 2D square lattice with free boundaries at the bond percolation thresholdpc=1/2. It is found that the probability ofkand more Incipient Spanning Clusters (ISC) have the valuesP(k>1) ≈ 0.00658(3)andP(k>2) ≈ 0.00000148(21)provided that the limit of these probabilities for infinite lattices exists. The probabilityP(k>3)of more than three ISC could be estimated to be of the order of10-11and 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 whenk≫1. We have detected a single sample with four ISC in a total number of about 1010samples 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.

Keywords: Monte Carlo Simulation; Coexistence of Spanning Clusters; Aizenman Theory (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1142/S0129183197000394

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