Papers › Percolation thresholds on high dimensional Dₙ and dense packing lattices
Percolation thresholds on high dimensional Dₙ and dense packing lattices
Yi Hu, Patrick Charbonneau
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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 Dₙ root lattices in n dimension as well as E₈-related dense packing lattices. Here, we consider the percolation problem on Dₙ for n=3 to $13$ and on E₈ 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 Dₙ 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.
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