Probability of Incipient Spanning Clusters in Critical Two-Dimensional Percolation

It was a common belief until a very recent time that on two-dimensional (2D) lattices at percolation threshold pc there exists exactly one percolation cluster [1,2]. New insight developed recently by Aizenman, who proved [3] that the number of Incipient Spanning Clusters (ISC) in 2D critical percolation can be larger than one, and that the probability of at least k separate clusters is bounded A e k 2 ≤ PL(k) ≤ e −α′ k , where α and α are constants and L is a linear lattice size. We investigate by Monte-Carlo the number of spanning clusters in the critical bond percolation model on 2D square lattices. Using simple finite-size scaling of probabilities on a selfdual lattices of moderate size, we have determined with a good accuracy values of the probabilities P (k) = limL→∞ PL(k) for k = 1, 2 and 3.