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

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.