Infinite volume Gibbs states of the generalized mean-field orthoplicial model

The generalized mean-field orthoplicial model is a particular type of mean-field model on a space of continuous spins on $\mathbb{R}^n$ that are constrained to a scaled $n-1$-dimensional $\ell_1$-sphere, equivalently a scaled $n-1$-dimensional orthoplex, and interact through a general interaction function. The finite volume Gibbs states of this model correspond to singular probability measures. In this paper, we rigorously classify the infinite volume Gibbs states of this model, and we show that they are convex combinations of product states. The primary purpose of this paper is to present the rigorous methods leading to the classification of the infinite volume Gibbs states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables.