Bond percolation in higher dimensions

We collect results for bond percolation on various lattices from two to fourteen dimensions that, in the limit of large dimension $d$ or number of neighbors $z$, smoothly approach a randomly diluted Erdős-Rényi graph. We include results on bond-diluted hypersphere packs in up to nine dimensions, which show the mean coordination, excess kurtosis, and skewness evolving smoothly with dimension towards the Erdős-Rényi limit.

Figure 1

Connected path in black across part of a sample with redundancy via the red (dark gray) loop, which is overconstrained with one redundant bond, and irrelevancy via the green (light gray) region, which contains dangling ends that are not involved in percolation.

Figure 2

Bond diluted finite Erdős-Rényi graph, where before dilution every node was connected to every other node.

Figure 3

Results for the mean coordination against the width for 2D lattices (red triangles), 3D lattices (blue squares), hypercubic lattices (black circles), and random hypersphere packings (gray circles) at the percolation threshold. The straight lines joining adjacent points are only for guidance of the eye. The thick green line is the Bethe lattice result with the isostatic point at (0,2) and the Erdős-Rényi result (1,1) shown as the large purple dot. The dashed line shows the result of a $1/(z-1)$ expansion [15] given in Eq. (11) for hypercubic lattices.

Figure 4

Skewness [thick red (dark gray) line] and excess kurtosis [thick green (light gray) line] as a function of the mean coordination $\langle r\rangle$ for Bethe lattices at the percolation threshold. Also shown are the skewness (triangles) and excess kurtosis (squares) for hypercubic lattices as gray symbols and random hypersphere packings as black symbols. The straight lines joining adjacent symbols are guides to the eye. The Erdős-Rényi result (1,1) is shown as the large purple dot. The dashed line shows the result of a $1/(z-1)$ expansion [15] given in Eq. (11).