On sampling symmetric Gibbs distributions on sparse random graphs and hypergraphs
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We consider efficient algorithms for approximate sampling from symmetric Gibbs distributions on the sparse random (hyper)graph. The examples we consider here include (but are not restricted to) important distributions on spin systems and spin-glasses such as the q state antiferromagnetic Potts model for q≥ 2, including the random 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 relies 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^2log n). It is evident that the algorithm requires a range of the parameters of the distributions that coincide with the tree-uniqueness region, parametrised w.r.t. the expected degree d. More precisely, this is true for distributions for which the uniqueness region is known. For many of the distributions we consider, we are far from establishing what is believed to be their uniqueness region. This imposes certain limitations to our purposes. We build a novel approach which utilises the notion of contiguity between Gibbs distributions and the so-called teacher-student model. With this approach we bring together tools and notions from sampling and statistical inference algorithms.
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