Approximate FKG inequalities for phase-bound spin systems, with applications to central limit theorems for exponential random graphs
Provides a theoretical foundation for statistical inference in exponential random graph models at low temperatures, addressing a known gap in the literature.
The authors prove that individual phases in exponential random graph models (ERGMs) satisfy an approximate FKG inequality, enabling central limit theorems within each phase in the phase-coexistence regime, answering a question by Bianchi, Collet, and Magnanini.
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.