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A set of lower bounds on the continuum percolation threshold $\eta_c$ ofoverlapping convex hyperparticles of general nonspherical (anisotropic) shapewith a specified orientational probability distribution in $d$-dimensionalEuclidean space have been derived [S. Torquato, J. Chem. Phys. {\bf 136},054106 (2012)]. The simplest of these lower bounds is given by $\eta_c \gev/v_{ex}$, where $v_{ex}$ is the $d$-dimensional exclusion volume of ahyperparticle and $v$ is its $d$-dimensional volume. In order to study theeffect of dimensionality on the threshold $\eta_c$ of overlapping nonsphericalconvex hyperparticles with random orientations here, we obtain a scalingrelation for $\eta_c$ that is based on this lower bound and a conjecture thathyperspheres provide the highest threshold among all convex hyperparticleshapes for any $d$. This scaling relation exploits the principle thatlow-dimensional continuum percolation behavior encodes high-dimensionalinformation. We derive a formula for the exclusion volume $v_{ex}$ of ahyperparticle in terms of its $d$-dimensional volume $v$, surface area $s$ and{\it radius of mean curvature} ${\bar R}$ (or, equivalently, {\it mean width}).These basic geometrical properties are computed for a wide variety ofnonspherical hyperparticle shapes with random orientations across alldimensions, including, among other shapes, various polygons for $d=2$, Platonicsolids, spherocylinders, parallepipeds and zero-volume plates for $d=3$ andtheir appropriate generalizations for $d \ge 4$. We then compute the lowerbound and scaling relation for $\eta_c$ for this comprehensive set of continuumpercolation models across dimensions. We show that the scaling relationprovides accurate {\it upper-bound} estimates of the threshold $\eta_c$ acrossdimensions and becomes increasingly accurate as the space $d$ increases.

Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles