Percolation thresholds for discorectangles: numerical estimation for a range of aspect ratios

TL;DR

This study uses Monte Carlo simulations to estimate percolation thresholds of discorectangles across various aspect ratios, revealing how thresholds compare with known shapes and approach rectangle values at high aspect ratios.

Contribution

It provides numerical estimates of percolation thresholds for discorectangles over a range of aspect ratios, filling a gap in the understanding of their percolation behavior.

Findings

Percolation thresholds for discs and sticks match known values.

Thresholds for intermediate aspect ratios lie between ellipse and rectangle thresholds.

Threshold approaches that of rectangles as aspect ratio increases.

Abstract

Using Monte Carlo simulation, we have studied the percolation of discorectangles. Also known as stadiums or two-dimensional spherocylinders, a discorectangle is a rectangle with semicircles at a pair of opposite sides. Scaling analysis was performed to obtain the percolation thresholds in the thermodynamic limits. We found: (i) for the two marginal aspect ratios $\varepsilon = 1$ (disc) and $\varepsilon \to \infty$ (stick) the percolation thresholds coincide with known values within the statistical error; (ii) for intermediate values of $\varepsilon$ the percolation threshold lies between the percolation thresholds for ellipses and rectangles and approaches the latter as the aspect ratio increases.