The continuum percolation threshold for interpenetrating squares and cubes

TL;DR

This study uses Monte Carlo simulations to accurately determine the percolation thresholds for interpenetrating squares and cubes, considering both aligned and randomly oriented configurations, revealing orientation-dependent critical fractions.

Contribution

It provides new precise estimates of percolation thresholds for squares and cubes in different orientations, filling gaps in existing data.

Findings

Aligned squares have a critical area fraction of 0.6666.

Randomly oriented squares have a critical area fraction of 0.6254.

Aligned cubes have a critical volume fraction of 0.2773.

Abstract

Monte Carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions. Simulations are performed for two cases: (i) objects whose edges are aligned parallel to one another and (ii) randomly oriented objects. For squares whose edges are aligned, the critical area fraction at the percolation threshold phi_c=0.6666 +/- 0.0004, while for randomly oriented squares phi_c=0.6254 +/- 0.0002, 6% smaller. For cubes whose edges are aligned, the critical volume fraction at the percolation threshold phi_c=0.2773 +/- 0.0002, while for randomly oriented cubes phi_c=0.2236 +/- 0.0002, 24% smaller.