Directed percolation: ‘‘field’’ exponents and a test of scaling in two and three dimensions

Directed percolation is a simple model which is believed to fall into the same universality class as Reggeon field theory. A large body of series data in both ‘‘field’’ and ‘‘thermal’’ (concentration) variables is now available for both of these models, and they provide an ideal framework for studying the relationship between field and thermal corrections to scaling. In particular it is of interest to test whether the measured nonanalytic confluent correction in the field direction, Ω, is equal to ${\Delta }_1$/βδ in any system. This relation between Ω and ${\Delta }_1$, the thermal correction and β, and 1/δ, the exponents of the percolation probability in the thermal and field directions, has been proposed but never successfully confirmed for isotropic percolation. We make direct estimates of Ω=0.45±0.15(2d), Ω=0.4±0.2(3d), 1/δ=0.111±0.003(2d), 1/δ=0.285±0.35(2d), and ${\Delta }_1$=0.75±0.10(3d), and invoke extant estimates of ${\Delta }_1$(2d) and β to confirm the scaling relation.