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N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let an $n\times n$ Matrix have entries which are either 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. Let $C_n$ be the total number of ``Site'' clusters. Then the value

K_S(p)=\lim_{n\to\infty} {\left\langle{C_n}\right\rangle{}\over n^2},
\end{displaymath} (1)

called the Mean Cluster Count Per Site or Mean Cluster Density, exists. Numerically, it is found that $K_S(1/2)\approx 0.065770\ldots$ (Ziff et al. 1997).

Considering instead ``Bond'' clusters (where numbers are assigned to the edges of a grid) and letting $C_n$ be the total number of bond clusters, then

K_B(p)\equiv \lim_{n\to\infty} {\left\langle{C_n}\right\rangle{}\over n^2}
\end{displaymath} (2)

exists. The analytic value is known for $p=1/2$,
K_B({\textstyle{1\over 2}})={\textstyle{3\over 2}}\sqrt{3}-{\textstyle{41\over 16}}
\end{displaymath} (3)

(Ziff et al. 1997).

See also Bond Percolation, Percolation Theory, s-Run, Site Percolation


Finch, S. ``Favorite Mathematical Constants.''

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.

© 1996-9 Eric W. Weisstein