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

TL;DR

This paper demonstrates that certain DLR states in the 3D Ising model at low temperature are not obtainable as limits of finite-volume measures with fixed boundary conditions, highlighting complexities in Gibbs state structures.

Contribution

It proves that the mixture of two reflection-symmetric Dobrushin states is a Gibbs state not arising from deterministic boundary conditions.

Findings

Mixture of Dobrushin states is a Gibbs state.

Such states are not limits of finite-volume measures with fixed boundary conditions.

Discussion on extending results to the Potts model.

Abstract

We review what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and in particular we prove that the mixture $\frac12(\mu^\pm+\mu^\mp)$ 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. Finally we point out what the issues are in order to extend the analysis to the Potts model, and give a few conjectures.