Effect of Dimensionality on the Continuum Percolation of Overlapping Hyperspheres and Hypercubes: II. Simulation Results and Analyses

摘要

In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136},054106 (2012)], analytical results concerning the continuum percolation ofoverlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$were obtained, including lower bounds on the percolation threshold. In thepresent investigation, we provide additional analytical results for certaincluster statistics, such as the concentration of $k$-mers and relatedquantities, and obtain an upper bound on the percolation threshold $\eta_c$. Weutilize the tightest lower bound obtained in the first paper to formulate anefficient simulation method, called the {\it rescaled-particle} algorithm, toestimate continuum percolation properties across many space dimensions withheretofore unattained accuracy. This simulation procedure is applied to computethe threshold $\eta_c$ and associated mean number of overlaps per particle${\cal N}_c$ for both overlapping hyperspheres and oriented hypercubes for $ 3\le d \le 11$. These simulations results are compared to corresponding upperand lower bounds on these percolation properties. We find that the boundsconverge to one another as the space dimension increases, but the lower boundprovides an excellent estimate of $\eta_c$ and ${\cal N}_c$, even forrelatively low dimensions. We confirm a prediction of the first paper in thisseries that low-dimensional percolation properties encode high-dimensionalinformation. We also show that the concentration of monomers dominate overconcentration values for higher-order clusters (dimers, trimers, etc.) as thespace dimension becomes large. Finally, we provide accurate analyticalestimates of the pair connectedness function and blocking function at theircontact values for any $d$ as a function of density.