Finitary Coding of the Sub-critical Ising Model with Finite Expected Coding Volume

It has been shown by van den Berg and Steif [3] that the sub-critical Ising model on Z is a finitary factor of a finite-valued i.i.d. process. We strengthen this by showing that the factor map can be made to have finite expected coding volume (in fact, stretched-exponential tails), answering a question of van den Berg and Steif [3]. The result holds at any temperature above the critical temperature. An analogous result holds for Markov random fields satisfying a high-noise assumption and for proper colorings with a large number of colors.