Percolation

Denoting by M(L) the number of cells in the largest cluster, simulations show in the limit of large L, M(L) scales as

ln(L)	for p < pc
Ld	for p = pc
L2	for p > pc

where d = 1.89 +/- 0.03.

That is, at the percolation threshold, the largest cluster is a statistically self-similar fractal.

For percolation on triangular lattices, in 1964 Sykes and Essam showed analytically that p_c = 1/2 and d = 91/48, and there is some evidence that this value of d is the same of all 2-dimensional lattices.

Return to percolation.