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 pc = 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.