Conformal invariance of double random currents and the XOR-Ising model II: tightness and properties in the discrete

This is the second of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising model on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents both with free or wired boundary conditions, and in the XOR-Ising models with free and plus/plus boundary conditions. Therefore we establish Wilsons conjecture on the XOR-Ising model. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the continuum Gaussian Free Field. In this paper, we derive crossing properties of the discrete models required to prove this characterization.