s-Cluster

Let a random  (0,1)-matrix have entries which are 1 (with probability ) or 0 (with probability ). An -cluster is an isolated group of  adjacent (i.e., horizontally or vertically connected) 1s. The counts of -clusters of various sizes are summarized in the following table for small  -matrices (OEIS A086266).

	number of -clusters for , 1, ...
1	1, 1
2	1, 13, 2
3	1, 218, 208, 78, 6, 1
4	1, 11506, 21172, 20262, 9560, 2593, 408, 32, 2

This gives the mean numbers of -clusters for , 2, ... as 1/2, 17/16, 897/512, 168529/65536, ... (OEIS A086265).

Let  be the total number of these "site" clusters. Then the value

called the mean cluster count per site or mean cluster density, exists. Numerically, it is found that

(OEIS A086268; Ziff et al. 1997).

REFERENCES

Finch, S. R. "Percolation Cluster Density Constants." §5.18 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 371-378, 2003.

Sloane, N. J. A. Sequences A086265, A086266, and A086268 in "The On-Line Encyclopedia of Integer Sequences."Temperley, H. N. V. and Lieb, E. H. "Relations Between the 'Percolation' and 'Colouring' Problem and Other Graph-Theoretical Problems Associated with Regular Planar Lattices; Some Exact Results for the 'Percolation' Problem." Proc. Roy. Soc. London A 322, 251-280, 1971.

Ziff, R. M.; Finch, S. R.; and Adamchik, V. S. "Universality of Finite-Sized Corrections to the Number of Critical Percolation Clusters." Phys. Rev. Let. 79, 3447-3450, 1997.