On the dependence of the zero-free region of a partition function on the external field

Let $\{0, 1\}^n$ be the Boolean cube, endowed with the probability product measure, where ${\Bbb P}(1)=p$ and ${\Bbb P}(0)=q$ with $0 < p \leq q=1-p$. Let $\phi_i: \{0, 1\}^n \longrightarrow {\Bbb C}$ be $1$-Lipschitz functions in the Hamming metric, such that each $\phi_i$ depends on at most $r$ coordinates of $x \in \{0, 1\}^n$, where $rp \geq 12$. For $j=1, \ldots, n$, let $I_j$ be the set of indices $i$ such that $\phi_i$ depends on the $j$-th coordinate. We prove that $E\thinspace \exp\left\{ \sum_{i=1}^m \lambda_i \phi_i \right\} \ne 0$ provided $\lambda_i \in {\Bbb C}$ satisfy $\sum_{i \in I_j} |\lambda_i| \leq {1 \over 10 \sqrt{rp}}$ for all $j$. This translates into a regime for $\pm 1$ spin systems, where a linear increase in the energy of multi-spin interactions requires only a logarithmic increase of the external field to keep the partition function zero-free and the system away from the phase transition.