### Abstract

We study the effect of dimensionality on the percolation threshold η_{c} of identical overlapping nonspherical convex hyperparticles in d-dimensional Euclidean space Rd. This is done by formulating a scaling relation for η_{c} that is based on a rigorous lower bound 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 R̄ (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.

Original language | English (US) |
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Article number | 022111 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 87 |

Issue number | 2 |

DOIs | |

State | Published - Feb 13 2013 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics