Universal Formulae for Percolation Thresholds

Abstract

A power law is postulated for both site and bond percolation thresholds. The formula writes $p_c=p_0[(d-1)(q-1){]}^{-a}{d}^b$, where $d$ is the space dimension and $q$ the coordination number. All thresholds up to $d\to \infty$ are found to belong to only three universality classes. For first two classes $b=0$ for site dilution while $b=a$ for bond dilution. The last one associated to high dimensions is characterized by $b=2a-1$ for both sites and bonds. Classes are defined by a set of value for $\{p_0;a\}$. Deviations from available numerical estimates at $d\le 7$ are within $\pm 0.008$ and $\pm 0.0004$ for high dimensional hypercubic expansions at $d\ge 8$. The formula is found to be also valid for Ising critical temperatures.

11 pages, latex, 3 figures not included

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