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The two common critical probabilities for a lattice graphLare the cluster size critical probabilitypH(L) and the mean cluster size critical probabilitypT(L). The values for the honeycomb latticeHand the triangular latticeTare proved to bepH(H) =pT(H) = 1–2 sin (π/18) andPH(T) =pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

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