Infinite-Volume Gibbs States of the Generalized Mean-Field Orthoplicial Model

Abstract

The generalized mean-field orthoplicial model is a 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 use probabilistic methods to rigorously classify the infinite-volume Gibbs states of this model, and we show that they are convex combinations of product 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.