Effect of dimensionality on the percolation threshold of overlapping nonspherical hyperparticles

We study the effect of dimensionality on the percolation threshold η_c of identical overlapping nonspherical convex hyperparticles in d-dimensional Euclidean space R^d. This is done by formulating a scaling relation for η_c that is based on a rigorous lower bound [Torquato, J. Chem. Phys.JCPSA60021-960610.1063/1.3679861 136, 054106 (2012)] and a conjecture that hyperspheres provide the highest threshold, for any d, among all convex hyperparticle shapes (that are not a trivial affine transformation of a hypersphere). This scaling relation also exploits the recently discovered principle that low-dimensional continuum percolation behavior encodes high-dimensional information. We derive an explicit formula for the exclusion volume v_ex of a hyperparticle of arbitrary shape in terms of its d-dimensional volume v, surface area s, and radius of mean curvature R¯ (or, equivalently, mean width). These basic geometrical properties are computed for a wide variety of nonspherical hyperparticle shapes with random orientations across all dimensions, including, among other shapes, various polygons for d=2, Platonic solids, spherocylinders, parallepipeds, and zero-volume plates for d=3 and their appropriate generalizations for d≥4. Using this information, we compute the lower bound and scaling relation for η_c for this comprehensive set of continuum percolation models across dimensions. We demonstrate that the scaling relation provides accurate upper-bound estimates of the threshold η_c across dimensions and becomes increasingly accurate as the space dimension increases.