Percolation thresholds and Fisher exponents in hypercubic lattices

Abstract

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