The LMP model and the Pirogov–Sinai strategy

Abstract

Systems of particles in the continuum are the object of this and of the next two chapters. In Sect. 10.1, some general considerations on the problems arising from such a new context are presented; in particular, it is shown that even the mean field case is non-trivial and cannot be explicitly solved. In Sect. 10.2 the LMP model (named after Lebowitz, Mazel and Presutti) is introduced and its mean field behavior established. In Sect. 10.3 the definition of contours and phase indicators of Chaps. 6 and 9 is adapted to the new context, and the main result about the validity of the Peierls bounds for a special, γ-dependent, value of the chemical potential is stated. In Sect. 10.4 a strategy of proof which follows the one used in Chap. 9 is outlined for the Ising case. It is seen that the absence of symmetry (the spin flip symmetry in the Ising model) leads to a new estimate involving the surface corrections to the pressure, which instead could be avoided in the Ising case. Such a new problem leads to a complete change of strategy which is outlined in Sect. 10.5. It is based on the Pirogov–Sinai theory in the Zahradnik version, which uses cutoff weights, with γ, the Kac scaling parameter, playing the role of the small temperatures in the classic Pirogov–Sinai models.

References can be found in Sect. 12.6.