On the zeros of partition functions with multi-spin interactions
Alexander Barvinok
Abstract
Let $X_1, \ldots, X_n$ be probability spaces, let $X$ be their direct product, let $φ_1, \ldots, φ_m: X \longrightarrow {\Bbb C}$ be random variables, each depending only on a few coordinates of a point $x=(x_1, \ldots, x_n)$, and let $f=φ_1 + \ldots + φ_m$. The expectation $E\thinspace e^{λf}$, where $λ\in {\Bbb C}$, 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 $φ_i$ is 1-Lipschitz in the Hamming metric of $X$, that each $φ_i(x)$ 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 $φ_i$ that depend on the coordinate $x_j$, we prove that $E\thinspace e^{λf} \ne 0$ provided $| λ| \leq \ (3 c \sqrt{r-1})^{-1}$ and that the bound is sharp up to a constant factor. Taking a scaling limit, we prove a similar result for functions $φ_1, \ldots, φ_m: {\Bbb R}^n \longrightarrow {\Bbb C}$ 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 $λ$ lies in a slightly smaller disc.