Percolation of discorectangles oriented predominantly along two mutually perpendicular directions

Abstract

The percolation of finite-thickness elongated objects---discorectangles (stadiums)---oriented predominantly along two mutually perpendicular directions is studied, extending recent results for zero-thickness sticks. The angular distribution is the ``Maltese cross'' parametrized by $α$. Based on the known analytic expression for the excluded area of a discorectangle pair, the percolation threshold is estimated by the eigenvalue method. The discorectangle connectivity matrix is represented as the stick matrix plus a rank-1 term determined by the constant part of the excluded area, $C=4lw+πw^{2}$, which gives the approximate relations $λ_{\max}^{\mathrm{DR}}\approxλ_{\max}^{\mathrm{stick}}+C$ and $n_c^{\mathrm{DR}}\approx n_c^{\mathrm{stick}}/(1+C n_c^{\mathrm{stick}})$. Finite thickness shifts the absolute thresholds and affects the first normalization, whereas the second normalization, written through maximum eigenvalues, cancels the additive geometric contribution and preserves only the orientational dependence, explaining the universality of the normalized curves. Monte Carlo simulations of unit-length discorectangles with aspect ratios 8, 16, and 32, various domain sizes, periodic boundary conditions, and union--find scaling analysis confirm the theoretical predictions.