Phase diagram of anisotropic planar Potts ferromagnets: a new conjecture

The exact phase diagram of the nearest-neighbour q-state Potts ferromagnet in the fully anisotropic 3-12 lattice is conjectured through a star-triangle transformation. It recovers all the available exact results concerning particular cases, namely: (i) anisotropic square lattice for all q; (ii) anisotropic triangular and honeycomb lattices for all q; (iii) anisotropic Kagome and diced lattices for q=2; (iv) isotropic 3-12 and Asanoha lattices for q=2. It provides proposals for several other planar lattices, in particular for the anisotropic Kagome (and diced) one for q not=2, where it differs slightly from the Wu(1979) conjecture (which also satisfies cases (i) and (iii)). The bond percolation critical probabilities on the 3-12 and Kagome lattices are determined to be respectively p c =0.739830 ... and p c =0.522372.