We show that for $q$-colorings in $k$-uniform hypergraphs with maximum degree $Δ$, if $k\ge 50$ and $q\ge 700Δ^{\frac{5}{k-10}}$, there is a "Lee-Yang" zero-free strip around the interval $[0,1]$ of the partition function, which includes the special case of uniform enumeration of hypergraph colorings. As an immediate consequence, we obtain Berry-Esseen type inequalities for hypergraph $q$-colorings under such conditions, demonstrating the asymptotic normality for the size of any color class in a uniformly random coloring. Our framework also extends to the study of "Fisher zeros", leading to deterministic algorithms for approximating the partition function in the zero-free region. Our approach is based on extending the recent work of [Liu, Wang, Yin, Yu, STOC 2025] to general constraint satisfaction problems (CSP). We focus on partition functions defined for CSPs by introducing external fields to the variables. A key component in our approach is a projection-lifting scheme, which enables us to essentially lift information percolation type analysis for Markov chains from the real line to the complex plane. Last but not least, we also show a Chebyshev-type inequality under the sampling LLL condition for atomic CSPs.

翻译：我们证明，对于最大度为Δ的k-均匀超图中的q-着色问题，当k≥50且q≥700Δ^{5/(k-10)}时，配分函数在区间[0,1]周围存在包含超图着色均匀计数特例的"李-杨"零自由带。作为直接推论，我们在此条件下获得了超图q-着色的Berry-Esseen型不等式，证明了均匀随机着色中任意颜色类大小的渐近正态性。我们的研究框架还可扩展至"费雪零点"的研究，从而在零自由区域内得到近似计算配分函数的确定性算法。本方法基于将[Liu, Wang, Yin, Yu, STOC 2025]的最新工作推广至一般约束满足问题。我们通过向变量引入外场，聚焦于CSP定义的配分函数。方法的核心是投影-提升方案，该方案使我们能够将马尔可夫链的信息渗流型分析从实轴本质性地提升到复平面。最后，我们还展示了原子CSP在采样LLL条件下的切比雪夫型不等式。