We introduce efficient algorithms for approximate sampling from symmetric Gibbs distributions on the sparse random (hyper)graph. The examples we consider include (but are not restricted to) important distributions on spin systems and spin-glasses such as the q state antiferromagnetic Potts model for $q\geq 2$, including the colourings, the uniform distributions over the Not-All-Equal solutions of random k-CNF formulas. Finally, we present an algorithm for sampling from the spin-glass distribution called the k-spin model. To our knowledge this is the first, rigorously analysed, efficient algorithm for spin-glasses which operates in a non trivial range of the parameters. Our approach builds on the one that was introduced in [Efthymiou: SODA 2012]. For a symmetric Gibbs distribution $\mu$ on a random (hyper)graph whose parameters are within an certain range, our algorithm has the following properties: with probability $1-o(1)$ over the input instances, it generates a configuration which is distributed within total variation distance $n^{-\Omega(1)}$ from $\mu$. The time complexity is $O((n\log n)^2)$. The algorithm requires a range of the parameters which, for the graph case, coincide with the tree-uniqueness region, parametrised w.r.t. the expected degree d. For the hypergraph case, where uniqueness is less restrictive, we go beyond uniqueness. Our approach utilises in a novel way the notion of contiguity between Gibbs distributions and the so-called teacher-student model.

翻译：本文提出了针对稀疏随机(超)图上对称吉布斯分布近似采样的高效算法。所考虑的实例包括（但不限于）自旋系统与自旋玻璃中的重要分布，例如$q\geq 2$的$q$态反铁磁波茨模型（含图着色问题），以及随机$k$-CNF公式的Not-All-Equal解均匀分布。此外，我们提出了一种名为$k$-自旋模型的自旋玻璃分布采样算法。据我们所知，这是首个在非平凡参数区间内严格可分析的自旋玻璃高效采样算法。本方法基于[Efthymiou: SODA 2012]提出的框架。对于参数处于特定范围内的随机(超)图上的对称吉布斯分布$\mu$，该算法具有以下性质：在输入实例上以概率$1-o(1)$生成一个配置，该配置与目标分布$\mu$的全变差距离为$n^{-\Omega(1)}$。时间复杂度为$O((n\log n)^2)$。算法所需的参数范围在图中对应于以期望度$d$参数化的树唯一性区域；对于超图情形（唯一性约束较弱），本方法可突破唯一性限制。本文创新性地利用吉布斯分布与所谓"师生模型"之间的邻接性概念。