Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes.

Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses.

In the first paper of this series [S. Torquato, J. Chem. Phys. 136, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in d-dimensional Euclidean space R(d) were obtained, including lower bounds on the percolation threshold. In the present investigation, we provide additional analytical results for certain cluster statistics, such as the concen...

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. 136, 054106 (2012)] and a conjecture that hyperspheres provide the highest threshold, for any d, among a...

Ja n 20 05 Possible crossover of a non universal quantity at the upper critical dimension

We report on a possible crossover of a non universal quantity at the upper critical dimensionality in the field of percolation. Plotting recent estimates for site percolation thresholds of hypercubes in dimension 6 ≤ d ≤ 13 against corresponding predictions from the GM formula pc = p0[(d − 1)(q − 1)] d for percolation thresholds, a significant departure of pc is observed for d ≥ 6. This result ...

Studies in statistical physics: Stochastic strategies in a minority game and continuum percolation of overlapping discs

1 2 D ec 1 99 6 Theory of continuum percolation III . Low density expansion

We use a previously introduced mapping between the continuum percolation model and the Potts fluid (a system of interacting s-states spins which are free to move in the continuum) to derive the low density expansion of the pair connectedness and the mean cluster size. We prove that given an adequate identification of functions, the result is equivalent to the density expansion derived from a co...