Constrained percolation, Ising model, and XOR Ising model on planar lattices

We study site percolation models on planar lattices including the [m,4,n,4] lattice and the square tilings on the Euclidean plane ( ) or the hyperbolic plane ( ), satisfying certain local constraints on degree‐4 faces. These models are closely related to Ising models and XOR Ising models (product of two i.i.d Ising models) on regular tilings of or . In particular, we obtain a description of the numbers of infinite “+” and “−” clusters of the ferromagnetic Ising model on a vertex‐transitive triangular tiling of for different boundary conditions and coupling constants. Our results show the possibility that such an Ising configuration has infinitely many infinite “+” and “−” clusters, while its random cluster representation has no infinite open clusters. Percolation properties of corresponding XOR Ising models are also discussed.