Decomposition of Binary Random Fields and Zeros of Partition Functions

Abstract

Let $\delta_c(X)$ denote the maximum $d$ in $\lbrack 0, \frac{1}{2}\rbrack$ such that a binary Gibbs random field $X$ can be decomposed as the modulo 2 sum of two independent binary fields, one of which is independent Bernoulli (white binary noise) of weight $d$. In a recent paper, Hajek and Berger showed, under modest assumptions, that $\delta_c > 0$. We point out here that the decomposition of $X$ is related to the classic statistical mechanics problem of determining zero-free regions of partition functions. A theorem of Ruelle is then applied to obtain improved estimates for $\delta_c$.