Universal Formulae for Percolation Thresholds

TL;DR

This paper proposes a universal power-law formula for percolation thresholds across dimensions, classifying them into three universality classes and validating the formula with numerical data and Ising critical temperatures.

Contribution

Introduces a universal formula for percolation thresholds that applies across dimensions and types, unifying thresholds and critical temperatures within a single framework.

Findings

The formula accurately predicts thresholds up to dimension 7.

Deviations from numerical estimates are within tight bounds.

The formula also applies to Ising critical temperatures.

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 $d\rightarrow \infty$ 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.