Bond Percolation — Square Lattice

Each bond on a square lattice is open with probability p, closed otherwise. At the critical threshold p_c = 1/2 (exact result, Kesten 1980), a spanning cluster first connects top to bottom with probability approaching 1 as N→∞. The order parameter P∞ (giant component fraction) vanishes as (p−p_c)^β with β=5/36. Watch the spanning cluster emerge near criticality.

0.50
p = 0.50 Spanning: P∞ = Clusters: Max cluster: