Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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. {\bf 136},
054106 (2012)], analytical results concerning the continuum percolation of
overlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$
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 concentration of $k$-mers and related
quantities, and obtain an upper bound on the percolation threshold $\eta_c$. We
utilize the tightest lower bound obtained in the first paper to formulate an
efficient simulation method, called the {\it rescaled-particle} algorithm, to
estimate continuum percolation properties across many space dimensions with
heretofore unattained accuracy. This simulation procedure is applied to compute
the 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 upper
and lower bounds on these percolation properties. We find that the bounds
converge to one another as the space dimension increases, but the lower bound
provides an excellent estimate of $\eta_c$ and ${\cal N}_c$, even for
relatively low dimensions. We confirm a prediction of the first paper in this
series that low-dimensional percolation properties encode high-dimensional
information. We also show that the concentration of monomers dominate over
concentration values for higher-order clusters (dimers, trimers, etc.) as the
space dimension becomes large. Finally, we provide accurate analytical
estimates of the pair connectedness function and blocking function at their
contact values for any $d$ as a function of density.

17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
2
The categories of this paper are: math
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th