Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

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.

7 pages, 1 table, 5 figures (1PS+4*Latex),uses epsf.sty Int.J.Mod.Phys. C (submitted to)

References in corpus (1)

• Probability distribution and sizes of spanning clusters at the percolation thresholds

Cited by in corpus (14)

• The Number of Incipient Spanning Clusters in Two-Dimensional Percolation
• Exact critical exponent for the shortest-path scaling function in percolation
• Universal Finite-Size Scaling for Percolation Theory in High Dimensions
• Geometric properties of the Fortuin-Kasteleyn representation of the Ising model
• Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters
• Critical amplitude ratios of the Baxter-Wu model
• Numerical results for crossing, spanning and wrapping in two-dimensional percolation
• On the Aizenman exponent in critical percolation
• Thermal Percolation for Interacting Monomers Adsorbed on Square Lattices
• Shape Effects of Finite-Size Scaling Functions for Anisotropic Three-Dimensional Ising Models
• Percolation of aligned rigid rods on two-dimensional triangular lattices
• Probability of Incipient Spanning Clusters in Critical Two-Dimensional Percolation
• Universal crossing probability in anisotropic systems
• Scale-invariant universal crossing probability in one-dimensional diffusion-limited coalescence