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Words in several independent Wigner matrices

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1957)

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.

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Correspondence to Alice Guionnet .

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© 2009 Springer-Verlag Berlin Heidelberg

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Guionnet, A. (2009). Words in several independent Wigner matrices. In: Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics(), vol 1957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69897-5_4

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