Finitary codings for gradient models and a new graphical representation for the six-vertex model

It is known that the Ising model on DOUBLE-STRUCK CAPITAL Zd at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction to being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume-order large deviation estimate for the energy. Results in the same spirit are shown for the Potts model, the so-called beach model, and the six-vertex model. We also introduce a coupling between the six-vertex model with c >= 2 and a new Edwards-Sokal type graphical representation of it, which we believe is of independent interest.