Percolation Threshold - Finite-Size Scaling & Critical Exponents

Lattice Size (L × L)

50 × 50

Site Percolation

Bond Percolation

Probability p

0.59 (p_c ≈ 0.593)

Color Clusters

Highlight Spanning Cluster

At critical threshold p_c, infinite cluster first appears:

Square lattice: p_c ≈ 0.593 (site)

Cluster size distribution at p_c:

n_s ~ s^{-τ}, τ ≈ 2.05

Correlation length divergence:

ξ ~ |p - p_c|^{-ν}, ν = 4/3

Empty

Occupied

Spanning

Percolation Prob.

Largest Cluster

Occupation p

0.59

Spanning Cluster

Num Clusters

Largest Cluster %

p = 0 p_c ≈ 0.593 p = 1

Critical Region

P(percolation): - Order parameter: -