Asymptotic Expansion of —The Schwinger–Dyson Equation Approach

Abstract

In the present chapter we develop all the necessary tools to prove the large-N asymptotic expansion for $\ln Z_N[V]$ up to $\mathrm {o}(1)$ terms, in the form described in (2.5.5). This asymptotic expansion contains N-dependent functionals of the equilibrium measure whose large-N asymptotic analysis will be carried out in Sections 6.1–6.3. We shall first obtain some a priori bounds on the fluctuations of linear statistics around their means computed versus the N-dependent equilibrium measure $\mu _{\mathrm {eq}}^{(N)}$. In other words, we consider observables given by integration against products of the centred measure: $\mathcal{L}_{N}^{(\varvec{\lambda })} = L_{N}^{(\varvec{\lambda })} - \mu _{\mathrm{eq}}^{(N)}$. Then we shall build on a bootstrap approach to the Schwinger–Dyson equations so as to improve these a priori bounds. We shall use these improved bounds so as to identify the leading and sub-leading terms in the Schwinger–Dyson equations what, eventually, leads to an analogue, at $\beta \not =1$, of the representation (2.5.5) which will be given in Proposition 3.3.6. Finally, upon integrating the relation (2.5.4) so as to to interpolate the partition function between a Gaussian and a general potential, we will get the N-dependent large-N asymptotic expansion of $\ln Z_N[V]$ in Proposition 3.4.1.