Ising systems with Kac potentials

Abstract

Chapter 9 is about the Ising model with no external magnetic field and with a ferromagnetic Kac interaction. The model is the same as the one considered in Chap. 4, the difference being that here, γ, the Kac scaling parameter, is kept small but fixed, while in Chap. 4 we have studied the Lebowitz–Penrose limit γ→0. The main results are described in Sect. 9.1. In Sect. 9.2 coarse grain variables and contours are defined following the analogous definitions of Chap. 6 for the L–P free energy functional. It is proved that (generalized) plus and minus boundary conditions produce in the thermodynamic limit different DLR states, if β is larger than the mean field critical value, d≥2, and γ correspondingly small. The proof uses the Peierls bounds on contours which are proved in Sect. 9.3.

References for the whole of Part III are in Sect. 12.6.