Cardy's Formula for some Dependent Percolation Models

We prove Cardy’s formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zero-temperature case of Domany’s stochastic Ising ferromagnet on the hexagonal lattice H (with alternating updates of two sublattices); it may also be realized on the triangular lattice T with flips when a site disagrees with six, five and sometimes four of its six neighbors.