We establish convergence to SLE6 of the Exploration Processes for the correlated bond-triangular type models studied by two of us in an earlier work (Chayes and Lei in Rev. Math. Phys. 19:511–565, 2007). This (rigorously) establishes that these models are in the same universality class as the standard site percolation model on the triangular lattice. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy’s Formula can be established for domains in this generality.

We show how to extract Cardy’s Formula for a general class of domains given the requisite interior analyticity statement. This is accomplished by a careful study of the interplay between discretization schemes and extraction of limiting boundary values. Of particular importance to the companion work (Binder et al. in J. Stat. Phys., 2010) we establish these results for slit domains and for the critical percolation models introduced in Chayes and Lei (Rev. Math. Phys. 19:511–565, 2007).