We study constrained percolation models on planar lattices including the $[m,4,n,4]$ lattice and the square tilings of the hyperbolic plane, satisfying certain local constraints on faces of degree 4, and investigate the existence of infinite clusters. The constrained percolation models on these lattices are closely related to Ising models and XOR Ising models on regular tilings of the Euclidean plane or the hyperbolic plane. In particular, we obtain a complete picture of the number of infinite "$+$" and "$-$" clusters of the ferromagnetic Ising model with the free boundary condition on a vertex-transitive triangular tiling of the hyperbolic plane with all the possible values of coupling constants. Our results show that for the Ising model on a vertex-transitive triangular tiling of the hyperbolic plane, it is possible that its random cluster representation has no infinite open clusters, while the Ising model has infinitely many infinite "$+$"-clusters and infinitely many infinite "$-$" 查看全文>>