Percolation thresholds and Fisher exponents in hypercubic lattices

We use invasion percolation to compute highly accurate numerical values for bond and site percolation thresholds p_c on the hypercubic lattice Z^d for d =4 ,...,13 . We also compute the Fisher exponent τ governing the cluster size distribution at criticality. Our results support the claim that the mean-field value τ =5 /2 holds for d ≥6 , with logarithmic corrections to power-law scaling at d =6 .