关于稀疏随机图和超图上对称吉布斯分布的采样
On sampling symmetric Gibbs distributions on sparse random graphs and hypergraphs
AI总结：
本文提出在稀疏随机图和超图上对对称吉布斯分布（如Potts模型、Not-All-Equal解及k-自旋模型）进行高效近似采样的算法，在非平凡参数范围内保证总变差距离$n^{-\Omega(1)}$，时间复杂度$O((n\log n)^2)$，并利用邻接概念超越唯一性区域。
摘要：
我们针对稀疏随机（超）图上的对称吉布斯分布，引入了高效的近似采样算法。我们考虑的示例包括（但不限于）自旋系统和自旋玻璃中的重要分布，例如 $q\geq 2$ 的 $q$ 态反铁磁 Potts 模型，包括着色问题，以及随机 $k$-CNF 公式的 Not-All-Equal 解的均匀分布。最后，我们提出了一种从称为 $k$-自旋模型的自旋玻璃分布中采样的算法。据我们所知，这是第一个在非平凡参数范围内运行的自旋玻璃的严格分析的高效算法。\n我们的方法建立在 [Efthymiou: SODA 2012] 中引入的方法之上。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 $\mu$，我们的算法具有以下性质：在输入实例上以 $1-o(1)$ 的概率，它生成一个与 $\mu$ 的总变差距离为 $n^{-\Omega(1)}$ 的配置。时间复杂度为 $O((n\log n)^2)$。\n该算法所需的参数范围，对于图的情况，与树唯一性区域一致，该区域以期望度数 $d$ 为参数。对于超图的情况，唯一性限制较少，我们超越了唯一性。我们的方法以一种新颖的方式利用了吉布斯分布与所谓的教师-学生模型之间的邻接概念。
英文摘要：
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 $μ$ 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 $n^{-Ω(1)}$ from $μ$. 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.
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