Abstract
Abstract We consider the problem of computing the partition function $\sum _x e^{f(x)}$ , where $f: \{-1, 1\}^n \longrightarrow {\mathbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$ . In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error $0 < \epsilon < 1$ in quasi-polynomial $n^{O(\ln n - \ln \epsilon )}$ time if the Lipschitz constant of the non-linear part of f with respect to the $\ell ^1$ metric on the Boolean cube does not exceed $1-\delta $ , for any $\delta>0$ , fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that $\sum _x e^{\tilde {f}(x)} \ne 0$ for complex-valued polynomials $\tilde {f}$ in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee–Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices.
Highlights
Introduction and main resultsThe Ising model is one of the oldest, most famous and most studied models in statistical physics; see [7] for a thorough introduction, description, results and references
We look at the computational complexity and complex zeros of the partition function in the Ising model
We extend the bounds of Theorem 1 to cubic polynomials on the Boolean cube {−1, 1}
Summary
The Ising model is one of the oldest, most famous and most studied models in statistical physics; see [7] for a thorough introduction, description, results and references. We look at the computational complexity and complex zeros of the partition function in the Ising model. This is a classical and currently very active area of research; see [6], [9], [10], [13], [16], [14], [15], [18], [19] and [22] for some recent results. Let {−1, 1} be the dimensional Boolean cube of all -vectors =
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