Main Results and Strategy of Proof

Abstract

In the first part of this chapter we gather the main results which follow from the analysis developed in this book. To start with, in Section 2.1, we discuss an example, in Theorem 2.1.1, of the leading large-N asymptotic expansion of $\ln \mathfrak {z}_N[W]$ where $\mathfrak {z}_N[W]$ is the unscaled partition function defined by (1.5.27). We shall also argue that the large-N asymptotic behaviour of (1.5.27)—whose integrand does not depend explicitly on N—can be deduced from the one of the rescaled model (2.5.1)—whose integrand depends explicitly on N—that we propose to study. Then, after presenting the per se model of interest and listing the assumptions on which our analysis builds in Section 2.2, we shall discuss the form of the large-N asymptotic expansion of the logarithm of the rescaled partition function $\ln Z_{N}[V]$ in Section 2.3. Then, in Section 2.4, we shall discuss the characterisation of the N-dependent equilibrium measure that is pertinent for our study as well as the form of the inverse $\mathcal {W}_N$ of a fundamental singular integral operator $\mathcal {S}_N$ that arises naturally in the study. Finally, Section 2.5, we outline the main steps of the proof.