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b-Cluster

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Let a random n×n (0,1)-matrix have entries which are 1 (with probability p) or 0 (with probability q=1-p) and numbers are assigned to the edges of a grid. A b-cluster is an isolated group of b adjacent (i.e., horizontally or vertically connected) 1s. Let C_n be the total number of "bond" clusters, then

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

exists. The analytic value is known for p=1/2,

K_B(1/2)=3/2sqrt(3)-(41)/(16)=0.0355762...

(OEIS A086269; Ziff et al. 1997).

See also

Bond Percolation, Connected Component, Percolation Theory, s-Cluster, 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. Sequence A086269 in "The On-Line Encyclopedia of Integer Sequences."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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b-Cluster

Cite this as:

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

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