We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice $$\mathbb {Z}^d$$ Z d and on the torus $$(\mathbb {Z}/n\mathbb {Z})^d$$ ( Z / n Z ) d . Our approach is based on combining contour representations from Pirogov–Sinai theory with Barvinok’s approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of $$\mathbb {Z}^d$$ Z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus $$(\mathbb {Z}/n\mathbb {Z})^d$$ ( Z / n Z ) d at sufficiently low temperature.

中文翻译：

---

算法 Pirogov-Sinai 理论

我们开发了一种有效的算法方法，用于在晶格 $$\mathbb {Z}^d$$ Z d 和环面 $$ 的有限子集上的一系列统计物理模型的低温状态下进行近似计数和采样(\mathbb {Z}/n\mathbb {Z})^d$$ ( Z / n Z ) d 。我们的方法基于将 Pirogov-Sinai 理论中的轮廓表示与 Barvinok 使用截断泰勒级数近似计数的方法相结合。我们的主要结果的一些结果包括 FPTAS 用于在具有适当边界条件的 $$\mathbb {Z}^d$$ Z d 的子集上以足够高的逸度逼近硬核模型的配分函数和有效的采样算法在足够低的温度下，离散环面 $$(\mathbb {Z}/n\mathbb {Z})^d$$ ( Z / n Z ) d 上的铁磁 Potts 模型。