Percolation thresholds on high dimensional $D_n$ and dense packing lattices

TL;DR

This study investigates percolation thresholds on high-dimensional $D_n$ and dense packing lattices, using simulations and series expansion, revealing trends and lattice-specific scaling behaviors.

Contribution

It extends percolation analysis to high-dimensional and dense packing lattices, providing precise thresholds and exploring finite-size scaling properties.

Findings

Bond percolation thresholds approach Bethe lattice limit with increasing dimension.

Finite-size scaling exponent varies with lattice type and percolation type.

Unexplained trends observed in correction terms for high-dimensional lattices.

Abstract

The site and bond percolation problems are conventionally studied on (hyper)cubic lattices, which afford straightforward numerical treatments. The recent implementation of efficient simulation algorithms for high-dimensional systems now also facilitates the study of $D_n$ root lattices in $n$ dimension as well as $E_8$-related dense packing lattices. Here, we consider the percolation problem on $D_n$ for $n=3$ to $13$ and on $E_8$ relatives for $n=6$ to 9. Precise estimates for both site and bond percolation thresholds obtained from invasion percolation simulations are compared with dimensional series expansion on $D_n$ lattices based on lattice animal enumeration. As expected, the bond percolation threshold rapidly approaches the Bethe lattice limit as $n$ increases for these high-connectivity lattices. Corrections, however, exhibit clear yet unexplained trends. Interestingly, the finite-size scaling exponent for invasion percolation is found to be lattice and percolation-type specific.