On the zeros of partition functions with multi-spin interactions

Let X_{1}, \ldots, X_{n} be probability spaces, let X be their direct product, let \phi_{1}, \ldots, \phi_{m}: X \rightarrow \mathbb{C} be random variables, each depending only on a few coordinates of x=(x_{1}, \ldots, x_{n}) , and let f=\phi_{1} + \cdots + \phi_{m} . The expectation \mathbf{E}\,e^{\lambda f} , where \lambda \in \mathbb{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 \phi_{i} is 1-Lipschitz in the Hamming metric of X , that each \phi_{i}(x) depends on at most r \geq 2 coordinates x_{1}, \ldots, x_{n} of x \in X , and that for each j there are at most c \geq 1 functions \phi_{i} that depend on the coordinate x_{j} , we prove that \mathbf{E}\,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 \phi_{1}, \ldots, \phi_{m}: \mathbb{R}^{n} \rightarrow \mathbb{C} that are 1-Lipschitz in the \ell^{1} metric of \mathbb{R}^{n} and where the expectation is taken with respect to the standard Gaussian measure in \mathbb{R}^{n} . As a corollary, the value of the expectation can be efficiently approximated, provided \lambda lies in a slightly smaller disc.