A wide class of problems in combinatorics, computer science and physics can be described along the following lines. There are a large number of variables ranging over a finite domain that interact through constraints that each bind a few variables and either encourage or discourage certain value combinations. Examples include the k-SAT problem or the Ising model. Such models naturally induce a Gibbs measure on the set of assignments, which is characterised by its partition function. The present paper deals with the partition function of problems where the interactions between variables and constraints are induced by a sparse random (hyper)graph. According to physics predictions, a generic recipe called the "replica symmetric cavity method" yields the correct value of the partition function if the underlying model enjoys certain properties [Krzkala et al., PNAS (2007) 10318-10323]. Guided by this conjecture, we prove general sufficient conditions for the success of the cavity method. The proofs are based on a "regularity lemma" for probability measures on sets of the form Ωn for a finite Ω and a large n that may be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 694-741, 2016.

中文翻译：

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利用Bethe的自由能。

组合语言学，计算机科学和物理学中的一类广泛的问题可以按照以下几行来描述。在有限域内有大量变量，它们通过约束相互影响，每个约束都绑定了几个变量，并鼓励或不鼓励某些值组合。示例包括k-SAT问题或Ising模型。这样的模型自然会在分配的集合上产生Gibbs测度，该测度的特征在于其分区功能。本文讨论了问题的分配函数，其中变量和约束之间的相互作用是由稀疏随机（超）图引起的。根据物理学预测，一种通用配方称为“复制对称腔法” 如果基础模型具有某些属性，则会产生分区函数的正确值[Krzkala等，PNAS（2007）10318-10323]。以此推测为指导，我们证明了腔法成功的一般充分条件。证明基于“正则引理”，用于对形式为ηn的有限Ω和大n的概率度量的概率度量，这些n可能具有独立的意义。©2016 Wiley Periodicals，Inc.随机结构。Alg。，49，694-741，2016。