First-order expansion

Abstract

At the end of this chapter (see Theorem 8.8) we will have proved that Lemma 7.12 holds as a first-order limit, i.e.,

$$\begin{array}{ll} \mathop {\lim }\limits_{N \to \infty } & \frac{1}{{N^2 }}\log \int {e\sum\nolimits_{i = 1}^n {t_i N{\rm Tr(}qi({\rm A}_{\rm 1} , \ldots ,{\rm A}_m ))d\mu ^N ({\rm A}_1 )} \cdots d\mu ^N ({\rm A}_m )} \\& =\sum\limits_{k_{\rm 1} , \cdots ,k_n \in N^n \backslash \{ 0\} } {\prod\limits_{i = 1}^n {\frac{{(t_i )^{k_i } }}{{k_i !}}M_0 ((q_i ,k_i ),1 \le i \le n)} } \\ \end{array}$$

provided the parameters t = (t_i)_1?i?n are sufficiently small and such that the polynomial V = ?t_iqi is strictly convex (i.e., belong to U_c ? B_? for some c > 0 and ? ? ?(c) for some ?(c) > 0). To prove this result we first show that, under the same assumptions, $\overrightarrow L _t^N (q) = \mu _{\sum {t_i q_i } }^N (N^{ - 1} {\rm Tr(}q{\rm )})$ converges as N goes to infinity to a limit that is as well related with map enumeration (see Theorem 8.4).

The central tool in our asymptotic analysis will be the so-called Schwinger- Dyson (or loop) equations. In finite dimension, they are simple emanation of the integration by parts formula (or, somewhat equivalently, of the symmetry of the Laplacian in L ²(dx)). As dimension goes to infinity, concentration inequalities show that $\overrightarrow L _t^N$ approximately satisfies a closed equation that we will simply refer to as the Schwinger-Dyson equation. The limit points of $\overrightarrow L _t^N$ will therefore satisfy this equation. We will then show that this equation has a unique solution in some small range of the parameters. As a consequence, $\overrightarrow L _t^N$ will converge to this unique solution. Showing that an appropriate generating function of maps also satisfies the same equation will allow us to determine the limit of $\overrightarrow L _t^N$.