The FKG inequality is an invaluable tool in monotone spin systems satisfying the FKG lattice condition, which provides positive correlations for all coordinate-wise increasing functions of spins. However, the FKG lattice condition is somewhat brittle and is not preserved when confining a spin system to a particular phase. For instance, consider the Curie-Weiss model, which is a model of a ferromagnet with two phases at low temperature corresponding to positive and negative overall magnetization. It is not a priori clear if each phase internally has positive correlations for increasing functions, or if the positive correlations in the model arise primarily from the global choice of positive or negative magnetization. In this article, we show that the individual phases do indeed satisfy an approximate form of the FKG inequality in a class of generalized higher-order Curie-Weiss models (including the standard Curie-Weiss model as a special case), as well as in ferromagnetic exponential random graph models (ERGMs). To cover both of these settings, we present a general result which allows for the derivation of such approximate FKG inequalities in a straightforward manner from inputs related to metastable mixing; we expect that this general result will be widely applicable. In addition, we derive some consequences of the approximate FKG inequality, including a version of a useful covariance inequality originally due to Newman as well as Bulinski and Shabanovich. We use this to extend the proof of the central limit theorem for ERGMs within a phase at low temperatures, due to the second author, to the non-forest phase-coexistence regime, answering a question posed by Bianchi, Collet, and Magnanini for the edge-triangle model.

翻译：FKG不等式是满足FKG格条件的单调自旋系统中不可或缺的工具，它为自旋的所有坐标递增函数提供了正相关性。然而，FKG格条件具有一定脆弱性，当将自旋系统限制在特定相时，该条件无法保持。例如，考虑Curie-Weiss模型，这是一个铁磁体模型，在低温下具有两个相，分别对应于正负总磁化强度。每个相内部是否对递增函数具有正相关性，或者模型中的正相关性是否主要源于全局正负磁化强度的选择，这并非先验明确的。本文证明，在一类广义高阶Curie-Weiss模型（包括标准Curie-Weiss模型作为特例）以及铁磁指数随机图模型（ERGMs）中，各个相确实满足近似形式的FKG不等式。为涵盖这两种情形，我们提出了一个通用结果，使得能够从与亚稳态混合相关的输入中直接推导此类近似FKG不等式；我们预期该通用结果将具有广泛适用性。此外，我们推导了近似FKG不等式的一些推论，包括最初由Newman以及Bulinski和Shabanovich提出的实用协方差不等式的变体。利用这一结果，我们将第二作者提出的低温下ERGM相内中心极限定理证明，推广至非森林相共存区域，从而回答了Bianchi、Collet和Magnanini针对边-三角模型所提出的问题。