Calculation of percolation thresholds in high dimensions for fcc, bcc, and diamond lattices

TL;DR

This paper extends the calculation of percolation thresholds to over forty high-dimensional lattices, revealing deviations from a proposed invariant and providing new data for fcc, bcc, and diamond lattices in 4 to 6 dimensions.

Contribution

It provides new percolation threshold values for various high-dimensional lattices and tests the validity of a proposed invariant, showing significant deviations in higher dimensions.

Findings

Deviations from the invariant up to 12% in 2D

Deviations increase to 69% in higher dimensions

New threshold values for fcc, bcc, and diamond lattices in 4-6D

Abstract

In a recent article, Galam and Mauger proposed an invariant for site and bond percolation thresholds, based on known values for twenty lattices (Eur. Phys. J. B 1 (1998) 255-258). Here we give a larger list of values for more than forty lattices in two to six dimensions. In this list are new results for fcc, bcc, and diamond lattices in 4, 5, and 6 dimensions. The list contains examples of lattices with equal site percolation thresholds, but different bond percolation thresholds. These and other examples show that there are deviations from the proposed invariant of up to 12% in two dimensions, increasing to 69% in higher dimensions.