On the zeros of partition functions with multi-spin interactions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909232207495168 |
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| author | Barvinok, Alexander |
| author_facet | Barvinok, Alexander |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_04179 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the zeros of partition functions with multi-spin interactions Barvinok, Alexander Probability Data Structures and Algorithms Mathematical Physics Combinatorics 82B20, 30C15, 68R05, 68W25 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. |
| title | On the zeros of partition functions with multi-spin interactions |
| topic | Probability Data Structures and Algorithms Mathematical Physics Combinatorics 82B20, 30C15, 68R05, 68W25 |
| url | https://arxiv.org/abs/2406.04179 |