We propose a family of ‘exactly solvable’ probability distributions to approximate partition functions of two-dimensional statistical mechanics models. While these distributions lie strictly outside the mean-field framework, their free energies can be computed in a time that scales linearly with the system size. This construction is based on a simple but nontrivial solution to the marginal problem. We formulate two non-linear constraints on the set of locally consistent marginal probabilities that simultaneously (i) ensure the existence of a consistent global probability distribution and (ii) lead to an exact expression for the maximum global entropy.

中文翻译：

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用于统计力学模型的完全可解的 ansatz

我们提出了一系列“完全可解”的概率分布来近似二维统计力学模型的配分函数。虽然这些分布严格位于平均场框架之外，但它们的自由能可以在与系统大小成线性比例的时间内计算出来。这种构造基于对边际问题的简单但非平凡的解决方案。我们在局部一致的边际概率集上制定了两个非线性约束，同时 (i) 确保一致的全局概率分布的存在和 (ii) 导致最大全局熵的精确表达式。