An Ergodic Description of Ground States

Given a translation-invariant Hamiltonian H, a ground state on the lattice Zd is a configuration whose energy, calculated with respect to H, cannot be lowered by altering its states on a finite number of sites. The set formed by these configurations is translation-invariant. Given an observable defined on the space of configurations, a minimizing measure is a translation-invariantprobability which minimizes the average of. If 0 is the mean contribution of all interactions to the site 0, we show that any configuration of the support of a minimizing measure is necessarily a ground state.