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

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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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Dates and versions

ensl-03053026 , version 1 (10-12-2020)

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  • HAL Id : ensl-03053026 , version 1

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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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