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：我们引入了高效的算法,从稀有随机(高率)图中随机吉布斯分布的对称分布中进行近似抽样。 我们认为的例子包括(但不限于)在旋转系统和旋转玻璃中的重要分布,例如$q\geq 2美元的q 州抗冬磁波模型。在随机的K-CNF公式的随机(高压)公式的随机(全价)解决方案中,我们引入了对称吉布斯分布中的对称分布。最后,我们展示了一种从旋转玻璃分布中取样的算法,称为k-spin模型。据我们所知,这是在非微不足道的参数范围内运行的对旋转玻璃系统的重要分布(但不限于) 。我们的方法建立在[Efthymiou:SODODA] 模型中引入的 。对于参数范围内的随机(超均匀) Gibsbs分布,我们的算法具有以下特性: 在独特输入实例中,它产生一种配置,在完全的模型距离内分配 $-\ Omegarr 时间范围, ialal deal case case case a.