s-Cluster
N.B. A detailed on-line essay by S. Finchwas the starting point for this entry.
Let an 
Matrix have entries which are either 1 (with probability 
) or 0 (with probability 
).An 
-cluster is an isolated group of adjacent (i.e., horizontally or vertically connected) 1s. Let 
be the total number of ``Site'' clusters. Then the value
 | (1) |
(Ziff et al. 1997).Considering instead ``Bond'' clusters (where numbers are assigned to the edges of a grid) and letting be thetotal number of bond clusters, then
 | (2) |
, | (3) |
References
Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/rndprc/rndprc.html 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.; Finch, S.; and Adamchik, V. ``Universality of Finite-Sized Corrections to the Number of Percolation Clusters.'' Phys. Rev. Let. To appear, 1998.
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