Percolation thresholds on high-dimensional Dn and E8 -related 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 $n$ dimensions as well as ${E}_{8}$-related 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 based on lattice animal enumeration for ${D}_{n}$ lattices. 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.