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Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

Lev N. Shchur·Sergey S. Kosyakov
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Abstract

The probability of simultaneous occurence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice at the bond percolation threshold $p_c=1/2$. It is found that the probability of k and more Incipient Spanning Clusters (ISC) has 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 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 a such value by nowdays 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 4 ISC in a total number of about 10^{10} samples investigated. The probability of single event is 1/10 for that number of samples.

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Lev N. Shchur, Sergey S. Kosyakov. 1997-02-27. Probability of Incipient Spanning Clusters in Critical Square Bond Percolation. https://doi.org/10.1142/s0129183197000394

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