Approximate FKG inequalities for phase-bound spin systems

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.