s-Cluster

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

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

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

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

$$K_B({\textstyle{1\over 2}})={\textstyle{3\over 2}}\sqrt{3}-{\textstyle{41\over 16}}$$ (3)

(Ziff et al. 1997).

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

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.