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s-Cluster


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

nnumber of s-clusters for s=0, 1, ...
11, 1
21, 13, 2
31, 218, 208, 78, 6, 1
41, 11506, 21172, 20262, 9560, 2593, 408, 32, 2

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

Let C_n(p) be the total number of these "site" clusters. Then the value

 K_S(p)=lim_(n->infty)(<C_n(p)>)/(n^2),

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

 K_S(1/2) approx 0.065770...

(OEIS A086268; Ziff et al. 1997).


See also

b-Cluster, Connected Component, Percolation Theory, s-Run, Site Percolation, Truchet Tiling

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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.

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s-Cluster

Cite this as:

Weisstein, Eric W. "s-Cluster." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/s-Cluster.html

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