DLR measures and the ergodic decomposition

Abstract

Chapter 12 capitalizes all information obtained from the analysis of Chap. 11. In Sect. 12.1 it is in fact proved that, as a consequence of the Peierls bounds, there are indeed at least two distinct DLR measures, the plus and minus DLR measures (obtained in the thermodynamic limit from Gibbs measures with generalized plus and minus boundary conditions). To see this requires some compactness properties of the set of Gibbs measures, which are considerably simpler than in general particle systems because of the special structure of the LMP hamiltonian. In Sect. 12.2 it is proved that the plus and minus DLR measures are unique (i.e. independent of the specific plus and minus boundary conditions used in their definition) and that they have a trivial σ-algebra at infinity (some more technical estimates are reported in a complementary section, Sect. 12.5). In Sect. 12.3 the classic Gallavotti–Miracle–Sole argument for the Ising model at low temperatures is adapted to the LMP model in order to show that any translation invariant DLR measure is a convex combination of the plus and minus DLR measures. In Sect. 12.4 it is finally shown that the plus and minus measures are translational invariant and ergodic (under the group of translations). The whole analysis in this chapter is to a large extent model independent once the Peierls bounds have been established.

Notes and References for the whole of Part III, including the present chapter, are in Sect. 12.6.