More on zeros and approximation of the Ising partition function
More on zeros and approximation of the Ising partition function
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DOI：
10.1017/fms.2021.40
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发表时间：
2020-05
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影响因子：
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通讯作者：
A. Barvinok;N. Barvinok
中科院分区：
文献类型：
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作者：
A. Barvinok;N. Barvinok
本文研究了配分函数$\sum _x e^{f（x）}$的计算问题，其中f：\{-1，1\}^n \longrightarrow {\mathbb R}$是布尔立方体$\{-1，1\}^n$上的二次或三次多项式.在一个二次多项式f的情况下，我们表明，分区函数可以近似在相对误差$0 0$，事先固定。对于一个三次多项式f，我们在一个更强的条件下得到同样的结果。我们应用多项式插值的方法，我们证明了$\sum _x e^{\tilde {f}（x）} \ne 0$对于复值多项式$\tilde {f}$在满足上述条件的实值f的邻域中。边界是渐近最优的。零自由区的结果被解释为在相应的伊辛模型的李-杨意义上的相变的情况下。边界的新特征是它们控制每个顶点的总交互，但不是顶点集的每个单个交互。
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 $0 0$ , 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.