Words in several independent Wigner matrices

In this chapter, we consider m independent Wigner N × N matrices ${\rm \{ A}^{{\rm N,l}} {\rm ,1} \le {\rm l} \le {\rm m\} }$ with real or complex entries. That is, the AN,l are self-adjoint random matrices with independent entries ${\rm (A}_{{\rm ij}}^{{\rm N,l}} ,1 \le i \le j \le N{\rm )}$ above the diagonal that are centered and with variance one. Moreover, the ${\rm (A}_{{\rm ij}}^{{\rm N,l}} ,1 \le i \le j \le N{\rm )}_{1 \le l \le m}$ are independent. We shall generalize Theorem 3.3 to the case where one considers words in several matrices, that is show that $N^{ - 1} Tr(A^{N,l_1 } A^{N,l_2 } \cdots A^{N,l_k } )$ converges for all choices of $l_i \in {\rm \{ 1, } \ldots {\rm , }m{\rm \} }$ and give a combinatorial interpretation of the limit. In Part VI, we describe the non-commutative framework proposed by D. Voiculescu to see the limit in the more natural framework of free probability. Here, we simply generalize Theorem 1.13 as a first step towards Part III. Let us first describe the combinatorial objects that we shall need.