We consider the problem of computing the partition function $\sum_x e^{f(x)}$, where $f: \{-1, 1\}^n \longrightarrow {\Bbb R}$ 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.
翻译:我们考虑的是计算分区函数$sum_x e ⁇ f(x)}美元的问题,其中美元为 $1, 1\n\n\ longrightrow ~Bb R} 美元是布尔兰立方体上美元=1, 1 ⁇ n美元的一个二次或立方多元值。 对于四边多元方美元, 我们显示分区函数可以在相对错误范围内大约为 $0 < \ epsilon < 1美元, 以准极价美元计 $n_O( ln- n- levilslon)} 美元时, 如果利普西茨非线性部分在布洛兰立方体立方立方体上美元=1美元, 而对于任何美元 delta >0美元, 之前固定的。 对于一个立方美元, 我们在一个比较坚固的状态下得到同样的结果。 我们采用多货币内部分析的方法, 如果利施法的功能是, 美元- 的正平面值区域 的正值 。