Let $X_1, \ldots, X_n$ be probability spaces, let $X$ be their direct product, let $\phi_1, \ldots, \phi_m: X \longrightarrow {\Bbb C}$ be random variables, each depending only on a few coordinates of a point $x=(x_1, \ldots, x_n)$, and let $f=\phi_1 + \ldots + \phi_m$. 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 $\phi_i$ 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 $E\thinspace e^{\lambda f} \ne 0$ 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 $\phi_1, \ldots, \phi_m: {\Bbb R}^n \longrightarrow {\Bbb C}$ 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.

翻译：令 $X_1, \ldots, X_n$ 为概率空间，$X$ 为其直积，$\phi_1, \ldots, \phi_m: X \longrightarrow {\Bbb C}$ 为随机变量，每个变量仅依赖于点 $x=(x_1, \ldots, x_n)$ 的少数坐标，且令 $f=\phi_1 + \ldots + \phi_m$。期望值 $E\thinspace e^{\lambda f}$（其中 $\lambda \in {\Bbb C}$）在统计物理学中表现为多自旋相互作用系统的配分函数，同时在组合数学与计算机科学中作为边着色模型、张量网络收缩或 Holant 多项式的配分函数出现。假设每个 $\phi_i$ 在 $X$ 的汉明度量下是 1-利普希茨的，每个 $\phi_i(x)$ 至多依赖于 $x \in X$ 的 $r \geq 2$ 个坐标 $x_1, \ldots, x_n$，且对每个 $j$ 至多有 $c \geq 1$ 个函数 $\phi_i$ 依赖于坐标 $x_j$，我们证明当 $| \lambda | \leq \ (3 c \sqrt{r-1})^{-1}$ 时 $E\thinspace e^{\lambda f} \ne 0$ 成立，且该界在常数因子范围内是尖锐的。通过尺度极限，我们对函数 $\phi_1, \ldots, \phi_m: {\Bbb R}^n \longrightarrow {\Bbb C}$ 证明了类似结论，这些函数在 ${\Bbb R}^n$ 的 $\ell^1$ 度量下是 1-利普希茨的，且期望值关于 ${\Bbb R}^n$ 中的标准高斯测度计算。作为推论，当 $\lambda$ 位于一个稍小的圆盘内时，该期望值可被高效逼近。