*N.B. A detailed on-line essay by S. Finch
was 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) |

Considering instead ``Bond'' clusters (where numbers are assigned to the edges of a grid) and letting be the
total 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.

© 1996-9

1999-05-26