In some cases, hexagons are used in percolation. Why do we use hexagons in percolation?
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$\begingroup$ math.stackexchange.com/questions/143979 $\endgroup$– Steve HuntsmanCommented Aug 14, 2017 at 12:11
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$\begingroup$ It's math... like all things, because coffee $\endgroup$– J...Commented Aug 15, 2017 at 2:08
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2 Answers
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Site percolation on the triangular lattice (or, equivalently, face percolation on the hexagonal lattice) is the only case for which conformal invariance of percolation at criticality has been proven. This has to do with very special combinatorial properties of the lattice. For a detailed explanation of this, see this paper by Vincent Beffara.
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The square lattice is self-dual, and it is of interest to study also lattices which are not self-dual. The critical probabilities for bond percolation of dual lattices satisfy $p_c=1-p'_c$, so the square lattice must have $p_c=1/2$, and one needs to go to a non-self-dual lattice for a nontrivial value. The triangular lattice is the most widely studied example [with $p_c=2\sin(\pi/18)]$, which is dual to the hexagonal lattice [with $p'_c=1-2\sin(\pi/18)]$.
answered Aug 14, 2017 at 10:19