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

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.