Bernoulli Percolation — Cluster Statistics near p_c

Bond percolation on a square lattice — critical phenomena

Each bond on the lattice is independently open with probability p. The percolation threshold p_c = 1/2 marks a second-order phase transition: below p_c, all clusters are finite; above p_c, an infinite spanning cluster exists (in the infinite system limit). Near p_c, the correlation length diverges as ξ ~ |p−p_c|^{−ν} with ν = 4/3, and cluster size distribution follows a power law n_s ~ s^{−τ} with τ = 187/91 ≈ 2.055 (Fisher exponent). Colors encode cluster identity — the largest cluster is shown in green.