We study algebraic properties of partition functions, particularly the location of zeros, through the lens of rapidly mixing Markov chains. The classical Lee-Yang program initiated the study of phase transitions via locating complex zeros of partition functions. Markov chains, besides serving as algorithms, have also been used to model physical processes tending to equilibrium. In many scenarios, rapid mixing of Markov chains coincides with the absence of phase transitions (complex zeros). Prior works have shown that the absence of phase transitions implies rapid mixing of Markov chains. We reveal a converse connection by lifting probabilistic tools for the analysis of Markov chains to study complex zeros of partition functions. Our motivating example is the independence polynomial on $k$-uniform hypergraphs, where the best-known zero-free regime has been significantly lagging behind the regime where we have rapidly mixing Markov chains for the underlying hypergraph independent sets. Specifically, the Glauber dynamics is known to mix rapidly on independent sets in a $k$-uniform hypergraph of maximum degree $\Delta$ provided that $\Delta \lesssim 2^{k/2}$. On the other hand, the best-known zero-freeness around the point $1$ of the independence polynomial on $k$-uniform hypergraphs requires $\Delta \le 5$, the same bound as on a graph. By introducing a complex extension of Markov chains, we lift an existing percolation argument to the complex plane, and show that if $\Delta \lesssim 2^{k/2}$, the Markov chain converges in a complex neighborhood, and the independence polynomial itself does not vanish in the same neighborhood. In the same regime, our result also implies central limit theorems for the size of a uniformly random independent set, and deterministic approximation algorithms for the number of hypergraph independent sets of size $k \le \alpha n$ for some constant $\alpha$.

翻译：我们通过快速混合的马尔可夫链视角研究配分函数的代数性质，特别是其零点分布。经典的Lee-Yang纲领通过定位配分函数的复零点开启了相变研究。马尔可夫链除了作为算法工具外，也被用于模拟趋于平衡的物理过程。在许多场景中，马尔可夫链的快速混合与相变（复零点）的缺失同时出现。先前研究已证明相变的缺失意味着马尔可夫链的快速混合。本文通过将分析马尔可夫链的概率工具提升至复平面以研究配分函数的复零点，揭示了逆向关联机制。我们的核心示例是$k$-均匀超图上的独立多项式，其中已知的最佳无零点区域远落后于对应超图独立集上存在快速混合马尔可夫链的区域。具体而言，已知Glauber动力学在最大度$\Delta$满足$\Delta \lesssim 2^{k/2}$的$k$-均匀超图的独立集上具有快速混合性；而$k$-均匀超图上独立多项式在点$1$附近的最佳已知无零点条件仅要求$\Delta \le 5$——这与普通图上的界相同。通过引入马尔可夫链的复延拓，我们将现有渗流论证提升至复平面，证明当$\Delta \lesssim 2^{k/2}$时，马尔可夫链在复邻域内收敛，且独立多项式在同一邻域内非零。在此条件下，我们的结果还推导出一致随机独立集规模的中心极限定理，以及对某些常数$\alpha$，规模为$k \le \alpha n$的超图独立集数量的确定性近似算法。