Examples of DLR states which are not weak limits of finite volume Gibbs measures with deterministic boundary conditions

We prove that the mixture 12 (µ ± + µ∓) of two reflection-symmetric Dobrushin states of the 3-dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finite-volume measures with deterministic boundary conditions. Furthermore, we discuss what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and give a few conjectures.