Monte Carlo simulations estimate site percolation threshold 0.822725 and bond threshold 0.798161 on the Smith hat tile, plus 0.544247 for site percolation on the dual graph.
P., Pichet, C., Pouliot, P
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Percolation Critical Probability of Aperiodic Smith Hat tile(1, $\sqrt3$)
Monte Carlo simulations estimate site percolation threshold 0.822725 and bond threshold 0.798161 on the Smith hat tile, plus 0.544247 for site percolation on the dual graph.