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) |
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.