Let a random (0,1)-matrix have entries which are 1 (with probability
) or 0 (with probability ). An -cluster is an isolated group of adjacent (i.e., horizontally or vertically connected) 1s.
The counts of -clusters
of various sizes are summarized in the following table for small -matrices (OEIS A086266).
number of -clusters for , 1, ...
1
1, 1
2
1,
13, 2
3
1, 218, 208,
78, 6, 1
4
1, 11506,
21172, 20262, 9560, 2593, 408, 32, 2
This gives the mean numbers of -clusters for , 2, ... as 1/2, 17/16, 897/512, 168529/65536, ... (OEIS
A086265).
Let be the total number of these "site" clusters. Then the value
called the mean cluster count per site or mean cluster density, exists. Numerically, it is found that
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. Sequences A086265,
A086266, and A086268
in "The On-Line Encyclopedia of Integer Sequences."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 A322, 251-280, 1971.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.
(0,1)-matrix have entries which are 1 (with probability
) or 0 (with probability 
). An 
-cluster is an isolated group of 
adjacent (i.e., horizontally or vertically connected) 1s.
The counts of 
-clusters
of various sizes are summarized in the following table for small 
-matrices (OEIS A086266).
-clusters for 
, 1, ...
-clusters for 
, 2, ... as 1/2, 17/16, 897/512, 168529/65536, ... (OEIS
A086265).
be the total number of these "site" clusters. Then the value
