On the zeros of partition functions with multi-spin interactions

Let $X1, \ldots, Xn$ be probability spaces, let $X$ be their direct product, let $\phi1, \ldots, \phim: X \longrightarrow {\Bbb C}$ be random variables, each depending only on a few coordinates of a point $x=(x1, \ldots, xn)$, and let $f=\phi1 + \ldots + \phim$. The expectation $E\thinspace e^{\lambda f}$, where $\lambda \in {\Bbb C}$, appears in statistical physics as the partition function of a system with multi-spin interactions, and also in combinatorics and computer science, where it is known as the partition function of edge-coloring models, tensor network contractions or a Holant polynomial. Assuming that each $\phii$ is 1-Lipschitz in the Hamming metric of $X$, that each $\phii(x)$ depends on at most $r \geq 2$ coordinates $x1, \ldots, xn$ of $x \in X$, and that for each $j$ there are at most $c \geq 1$ functions $\phii$ that depend on the coordinate $xj$, we prove that $E\thinspace e^{\lambda f} \ne 0$ provided $| \lambda | \leq \ (3 c \sqrt{r-1})^{-1}$ and that the bound is sharp up to a constant factor. Taking a scaling limit, we prove a similar result for functions $\phi1, \ldots, \phim: {\Bbb R}ⁿ \longrightarrow {\Bbb C}$ that are 1-Lipschitz in the $\ell^1$ metric of ${\Bbb R}^n$ and where the expectation is taken with respect to the standard Gaussian measure in ${\Bbb R}^n$. As a corollary, the value of the expectation can be efficiently approximated, provided $\lambda$ lies in a slightly smaller disc.