Asymptotic Analysis of Integrals

Abstract

In this chapter we carry out the large-N asymptotic analysis of the single and double integrals that arise in the problem. First, in Section 6.1.1, we deal with the one-fold integrals that arise in the characterisation of the image space $\mathfrak {X}_{s}(\mathbb {R})$ of $H_{s}\big ( [ a_N \,; b_N ] \big )$ under the operator $\mathcal {S}_N$. Then, in Section 6.1.2, we evaluate asymptotically in N one-dimensional integrals of $\mathcal {W}_N[H]$ versus test functions G. This provides the first set of results that were necessary in Section 3.4 for a thorough calculation of the large-N expansion of the partition function. Then, in Section 6.2, we build on the obtained large-N expansion of the two types of single integrals so as to characterise the support of the equilibrium measure. Finally, in Section 6.3, we obtain the large-N expansion, up to a vanishing with N remainder, of the double integral (3.4.3) arising in the large-N expansion of the partition function at $\beta =1$.