Abstract

In this chapter we deal with the large-volume limit, often called the thermodynamic limit. While the problem has received a lot of attention in deterministic systems from the early 1960s both in statistical mechanics (Ruelle (1999)) and Euclidean quantum field theory (Guerra (1972)), in random systems a major breakthrough has been the introduction of the quadratic interpolation method by Guerra and Toninelli (2002). We review the results for finite-dimensional systems with Gaussian or,more generally, centered interactions.We then extend the analysis to quantum models and to non-centered interactions satisfying a thermodynamic stability condition. Special attention is devoted to the correction to the leading term, i.e. the surface pressure, which is investigated for various boundary conditions and for a wide class of models. A complete result is obtained on the Nishimori line where we can make use of the full correlation inequalities set introduced in the previous chapter. Finally the mean-field case is analyzed for the relevant models that appear in the literature.

Introduction

The infinite-volume limit in spin glasses has been tackled for some time (Vuillermot (1977); Ledrappier (1977); Pastur and Figotin (1978); Khanin and Sinai (1979); van Enter and van Hemmen (1983); Zegarlinski (1991)). The difficulties with respect to the deterministic case arose due to the randomness of the interaction which requires both the study of the averaged quantities as well as the random ones. For quantities like pressure (free energy) and ground state energy per particle, the fluctuation between samples vanishes for large volumes (self-averaging), therefore it suffices to study the limit of the average value.