Universal Formulae for Percolation Thresholds

Details Universal Formulae for Percolation Thresholds Universal Formulae for Percolation Thresholds

1996-06-12

arxiv.org/abs/cond-mat/9606081v1

Phys. Rev. E 53, 2177 (1996)

Physics

Condensed Matter

11 pages, latex, 3 figures not included

Scientific paper

10.1103/PhysRevE.53.2177

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 $d\rightarrow \infty$ 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 $d \leq 7$ are within $\pm 0.008$ and $\pm 0.0004$ for high dimensional hypercubic expansions at $d \geq 8$. The formula is found to be also valid for Ising critical temperatures.

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