Universal formulas for percolation thresholds

A power law is postulated for both site and bond percolation thresholds. The formula is ${\mathit{p}}_{\mathit{c}}$=${\mathit{p}}_0$[(d-1)(q-1)${]}^{\mathrm{-}\mathit{a}}$ ${\mathit{d}}^{\mathit{b}}$, where d is the space dimension and q the coordination number. All thresholds up to d→∞ are found to belong to only three universality classes. For two classes b=0 for site dilution while b=a for bond dilution. The remaining class associated with high dimensions is characterized by b=2a-1 for both sites and bonds. Classes are defined by a set of value for {${\mathit{p}}_0$;a}. Deviations from available numerical estimates at d≤7 are within ±0.008 and ±0.0004 for high dimensional hypercubic expansions at d≥8. The formula is found to be also valid for Ising critical temperatures. © 1996 The American Physical Society.