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On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices

Abstract : Let X^N = (X^N_1 , ... , X^N_d) be a d-tuple of N × N independent GUE random matrices and Z^{NM} be any family of deterministic matrices in M_N(C) ⊗ M_M (C). Let P be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of P(X^N) converges towards a deterministic measure defined thanks to free probability theory. Let now f be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of 1/(MN) Tr ⊗ Tr( f(P (X^N ⊗ I_M , Z^{NM}) ) , and its limit when N goes to infinity. If f is six times differentiable, we show that it is of order M^2 N^{−2} and we compute an explicit upper bound of the difference. As a corollary we obtain a new proof and slightly improve a result of Haagerup and Thorbjørnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in (X^N , Z^{NM}, Z^{NM}*) to converge almost surely towards its free limit.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-03049388
Contributor : Felix Parraud <>
Submitted on : Wednesday, December 9, 2020 - 7:04:47 PM
Last modification on : Tuesday, December 15, 2020 - 3:32:38 AM
Long-term archiving on: : Wednesday, March 10, 2021 - 8:01:29 PM

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  • HAL Id : ensl-03049388, version 1

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Benoît Collins, Alice Guionnet, Félix Parraud. On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices. 2020. ⟨ensl-03049388⟩

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