Second-order expansion for the free energy

Abstract

At the end of this chapter, we will have proved that Lemma 7.12 holds, up to the second-order correction in the large N limit, i.e., that

$$\begin{array}{ll} & {\rm }\frac{1}{{N^2 }}\log \left( {\int {e^{\sum\nolimits_{i = 1}^n {tiN{\rm Tr}(qi(X_1 , \ldots ,X_m ){\rm )}} } d\mu ^N (X_1 ) \ldots d\mu ^N (X_m )} } \right) \\ = & \sum\limits_{g = 0}^1 {\frac{1}{{N^{2g - 2} }}\sum\limits_{k_1 , \ldots ,kn \in N} {\prod\limits_{i = 1}^n {\frac{{(t_i )^{K_i } }}{{k_i !}}M_g ((q_i ,k_i ),1 \le i \le n) + o\left( {\frac{1}{{N^2 }}} \right)} } } \\ \end{array}$$

when the parameters ti are small enough and such that ?t_iq_i is c-convex. As for the first order, we shall prove first a similar large N expansion for $\overrightarrow L _t^N$. We will first refine the arguments of the proof of Theorem 8.3 to estimate $(\overrightarrow L _t^N - T_t )$. This will already prove that $(\overrightarrow L _t^N - T_t )(P)$ is at most of order N_?2 for any polynomial P. To get the limit of $N^2(\overrightarrow L _t^N - T_t )(P)$, we will first obtain a central limit theorem for $(\mathop {L^N }\limits^ - T_t )$ which is of independent interest. The key argument in our approach, besides further uses of integration by partslike arguments, will be the inversion of a differential operator acting on noncommutative polynomials which can be thought as a non-commutative analog of a Laplacian operator with a drift.