The Fortuin-Kasteleyn-Ginibre (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. This inequality has numerous applications and plays an integral role in the proof of various central limit theorems (CLTs), including recent work on ferromagnetic exponential random graph models (ERGMs) wherein a Hamiltonian tilt promotes the presence of small subgraphs like triangles. However, the FKG lattice condition fails to hold when confining a spin system to a particular phase in the low-temperature regime of parameters. Thus it is not a priori clear if each phase internally has positive correlations for increasing functions, or if the positive correlations in the overall model (which is a mixture of phases) arise primarily from the global choice of phase. In this article, we show that the individual phases in ERGMs do indeed satisfy an approximate form of the FKG inequality internally. We use this to finish the proof of various CLTs within each individual phase in the phase-coexistence regime, answering a question posed by Bianchi, Collet, and Magnanini. We present the FKG inequality for ERGMs as a consequence of a more general result which holds under certain inputs related to metastable mixing; we expect this general result to be widely applicable, and we devote a section to spelling out the details of its application to a class of generalized higher-order ferromagnetic Curie-Weiss models where the necessary inputs are relatively transparent.

翻译：Fortuin-Kasteleyn-Ginibre (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，它为所有坐标递增的自旋函数提供了正相关性。该不等式具有众多应用，并在多种中心极限定理（CLT）的证明中发挥关键作用，包括近期关于铁磁指数随机图模型（ERGM）的研究——其中哈密顿倾斜促进了三角形等小子图的存在。然而，当自旋系统被约束在低温参数区域的特定相中时，FKG格条件不再成立。因此，我们无法先验地确定每个相内部对递增函数是否具有正相关性，或整个模型（作为相的混合）中的正相关性是否主要源于全局的相选择。本文证明了ERGM中的单个相确实在内部满足FKG不等式的近似形式。我们利用此结果完成了相共存区域内每个相内部多种中心极限定理的证明，回答了Bianchi、Collet和Magnanini提出的问题。我们将ERGM的FKG不等式作为更一般结论的推论提出，该结论在满足与亚稳态混合相关的特定输入条件下成立；我们预期这一一般结论具有广泛适用性，并专辟一节详细阐述其在广义高阶铁磁Curie-Weiss模型类中的应用，其中所需的输入条件相对透明。