Finitary coding for the sub-critical Ising model with finite expected coding volume

TL;DR

This paper proves that the sub-critical Ising model on integer lattices can be represented as a finitary factor of an i.i.d. process with finite expected coding volume, extending previous results and applying to related models.

Contribution

It establishes that the factor map for the sub-critical Ising model has finite expected coding volume with stretched-exponential tails, answering a key open question.

Findings

Finitary coding map has finite expected volume with stretched-exponential tails.

Results hold for all temperatures above the critical point.

Analogous results apply to high-noise Markov random fields and large-color proper colorings.

Abstract

It has been shown by van den Berg and Steif that the sub-critical Ising model on $\mathbb{Z}^d$ 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. 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.