Constrained Percolation on Z2

We study a constrained percolation process on Z, and prove the almost sure nonexistence of infinite clusters and contours for a large class of probability measures. Unlike the unconstrained case, in the constrained case no stochastic monotonicity is known. We prove the nonexistence of infinite clusters and contours for the constrained percolation by developing new combinatorial techniques which make use of the planar duality and symmetry. Applications include the almost sure nonexistence of infinite homogeneous clusters for the critical dimer model on the square-octagon lattice, as well as the almost sure nonexistence of infinite monochromatic contours and infinite clusters for the critical XOR Ising model on the square grid. By relaxing the symmetric condition of the underlying Gibbs measure on the constrained percolation process, we prove that there exists at most one infinite monochromatic contour for the non-critical XOR Ising model.