Directed percolation in four dimensions is of direct physical relevance to the world with three space and one time dimension. We present a comprehensive analysis of recently extended series for the moments of the cluster-size distribution and for the percolation probability in a ‘‘field’’ on the hypercubic lattice. We find a critical threshold, $p_c$=0.3025±0.0010, and dominant critical exponents γ=1.21±0.05, for the mean cluster size; β=0.82±0.03 and 1/δ=0.45±0.02 for the percolation probability in the thermal and field directions respectively; and a gap exponent of Δ=2.03±0.06. We find a thermal-correction exponent ${\Delta }_1$=0.55±0.15 and a field correction of Ω=0.3±0.1. We also calculate some universal critical-amplitude ratios.

• Received 11 December 1987

DOI:https://doi.org/10.1103/PhysRevB.37.7529