Percolation thresholds of two-dimensional continuum systems of rectangles [with erratum]

TL;DR

This paper introduces an efficient Monte Carlo method to accurately determine percolation thresholds in 2D systems of randomly-oriented rectangles, confirming and extending the excluded area theory with generalized predictive formulas.

Contribution

It provides high-precision percolation thresholds for various rectangle systems and extends the excluded area theory with new predictive formulas.

Findings

Percolation thresholds are mainly determined by average excluded areas.

The study confirms the validity of the excluded area theory for most systems.

Generalized formulas effectively predict percolation thresholds across diverse rectangle configurations.

Abstract

The present work introduces an efficient Monte Carlo algorithm for continuum percolation composed of randomly-oriented rectangles. By conducting extensive simulations, we report high precision percolation thresholds for a variety of homogeneous systems with different rectangle aspect ratios. This work verifies and extends the excluded area theory. It is confirmed that percolation thresholds are dominated by the average excluded areas for both homogeneous and heterogeneous rectangle systems (except for some special heterogeneous systems where the rectangle lengths differ too much from one another). In terms of the excluded areas, generalized formulae are proposed to effectively predict precise percolation thresholds for all these rectangle systems. This work is therefore helpful for both practical applications and theoretical studies concerning relevant systems. The Erratum addresses errors in our earlier paper "Percolation thresholds of two-dimensional continuum systems of rectangles" published in [Phys. Rev. E 88, 012101 (2013)].