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