一种用于复制对称边际的邻接方法
A contiguity approach to replica symmetric marginals
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中文总结 AI 辅助
本文提出概率腔邻接框架，以高温谢林顿-柯克帕特里克模型为例证明平均场吉布斯系统局部边际的复制对称收敛性，该方法可推广至其他平均场吉布斯系统。
中文摘要 AI 辅助
我们开发了一种概率腔邻接框架，用于证明平均场吉布斯系统中局部边际的复制对称收敛性。该方法基于哈密顿量的腔分解，结合通过拉东-尼科迪姆导数和赫林格型估计对概率测度进行直接比较。在概念层面，该方法将相关序参数的集中性与渐近腔模型的识别及相关吉布斯测度的分离开来。与基于插值的方法不同，该论证对无序的高斯性依赖较弱，且自然适配庞加莱和对数索伯列夫不等式等集中工具。为使方法清晰易懂，我们未追求最大通用性，而是在一个典型例子中实现了该框架：高温谢林顿-柯克帕特里克模型。在该设定下，我们证明固定自旋的边际律在全变差意义下收敛到复制方法预测的有效一维腔测度。除谢林顿-柯克帕特里克模型的特定结果外，本文还阐述了更广泛的腔邻接方法论，预计该方法可自然推广到其他平均场吉布斯系统，尤其是存在或不存在失配的贝叶斯推理模型。
英文摘要
We develop a probabilistic cavity-contiguity framework for proving replica-symmetric convergence of local marginals in mean-field Gibbs systems. The approach is based on cavity decompositions of the Hamiltonian together with direct comparison of probability measures through Radon-Nikodym derivatives and Hellinger-type estimates. At a conceptual level, the method separates the concentration of the relevant order parameters from the identification of the asymptotic cavity model and the comparison of the associated Gibbs measures. In contrast with interpolation-based approaches, the argument relies only weakly on the Gaussianity of the disorder and naturally accommodates concentration tools such as Poincaré and log-Sobolev inequalities. Rather than pursuing maximal generality, with the aim of making the method transparent, we implement the framework in a canonical example: the high-temperature Sherrington-Kirkpatrick model. In this setting, we prove that the marginal law of a fixed spin converges in total variation toward the effective one-dimensional cavity measure predicted by the replica method. Beyond the specific result for the Sherrington-Kirkpatrick model, the paper illustrates a broader cavity-contiguity methodology which is expected to extend naturally to other mean-field Gibbs systems, particularly Bayesian inference models with or without mismatch.
