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Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

Author

Listed:
  • Lev N. Shchur

    (Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia)

  • Sergey S. Kosyakov

    (Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia)

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.

Suggested Citation

  • Lev N. Shchur & Sergey S. Kosyakov, 1997. "Probability of Incipient Spanning Clusters in Critical Square Bond Percolation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 8(03), pages 473-481.
    • Handle: RePEc:wsi:ijmpcx:v:08:y:1997:i:03:n:s0129183197000394
    DOI: 10.1142/S0129183197000394
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