Directed Percolation — Nonequilibrium Phase Transition

ρ(t) ~ (p−p_c)^β · t^(−δ) — DP universality class, p_c ≈ 0.6447 (1+1D)

Occupation prob p 0.644

Lattice width L 150

Boundary

Stochastic noise 0.0

Steps/frame 3

Color mode

Directed percolation (DP) is the paradigmatic nonequilibrium phase transition to an absorbing state. In 1+1D: p_c ≈ 0.6447. Critical exponents: β ≈ 0.277 (order parameter), ν_∥ ≈ 1.734, ν_⊥ ≈ 1.097, δ ≈ 0.159 (density decay). Below p_c: all clusters die (absorbing phase). Above p_c: active phase persists. Reggeon field theory (Grassberger-de la Torre 1979): field-theoretic description; DP critical point is the same fixed point. DP conjecture: any model with a scalar order parameter, short-range interactions, no extra symmetry, and a single absorbing state belongs to the DP universality class.