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.

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更多关于 Ising 配分函数的零点和近似值

我们考虑计算配分函数的问题 $\sum _x e^{f(x)}$ ， 在哪里 $f: \{-1, 1\}^n \longrightarrow {\mathbb R}$ 是布尔立方体上的二次或三次多项式 $\{-1, 1\}^n$ . 在二次多项式的情况下F，我们证明了配分函数可以在相对误差范围内近似 $0 < \epsilon < 1$ 拟多项式 $n^{O(\ln n - \ln \epsilon )}$ 时间，如果非线性部分的 Lipschitz 常数F相对于该 $\ell ^1$ 布尔多维数据集上的度量不超过 $1-\delta $ , 对于任何 $\delta>0$ ，提前修复。对于三次多项式F，我们在稍强的条件下得到相同的结果。我们应用多项式插值的方法，为此我们证明 $\sum _x e^{\波浪号 {f}(x)} \ne 0$ 对于复值多项式 $\波浪号 {f}$ 在一个实际值附近F满足上述条件。边界是渐近最优的。零自由区域的结果被解释为在相应的 Ising 模型中不存在 Lee-Yang 意义上的相变。边界的新颖特征是它们控制每个顶点的总交互，而不是顶点集的每个单个交互。