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Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

Abstract : Let X_N be a family of N × N independent GUE random matrices, Z_N a family of deterministic matrices, P a self-adjoint non-commutative polynomial, that is for any N , P(X_N) is self-adjoint, f a smooth function. We prove that for any k, if f is smooth enough, there exist deterministic constants α_P^i (f, Z_N) such that E[1/N Tr f (P(X_N , Z_N)) ] = \sum^k_{ i=0} α_P^i (f, Z N) N^{-2i} + O(N^{−2k−2}). Besides the constants α_P^i (f, Z_N) are built explicitly with the help of free probability. In particular, if x is a free semicircular system, then when the support of f and the spectrum of P (x, Z_N) are disjoint, for any i, α_P^i (f, Z_N) = 0. As a corollary, we prove that given α < 1/2, for N large enough, every eigenvalue of P(X_N , Z_N) is N^{−α}-close from the spectrum of P(x, Z_N).
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-03053026
Contributor : Felix Parraud <>
Submitted on : Thursday, December 10, 2020 - 8:46:31 PM
Last modification on : Monday, March 29, 2021 - 2:46:58 PM
Long-term archiving on: : Thursday, March 11, 2021 - 9:08:52 PM

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Félix Parraud. Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices. 2020. ⟨ensl-03053026⟩

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