https://hal-ens-lyon.archives-ouvertes.fr/ensl-03049388Collins, BenoîtBenoîtCollinsKyoto - Department of Mathematics, Kyoto University - Kyoto University [Kyoto]Guionnet, AliceAliceGuionnetParraud, FélixFélixParraudUMPA-ENSL - Unité de Mathématiques Pures et Appliquées - ENS Lyon - École normale supérieure - Lyon - CNRS - Centre National de la Recherche ScientifiqueKyoto - Department of Mathematics, Kyoto University - Kyoto University [Kyoto]On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matricesHAL CCSD2022[MATH.MATH-PR] Mathematics [math]/Probability [math.PR][MATH.MATH-OA] Mathematics [math]/Operator Algebras [math.OA]Parraud, Felix2020-12-09 19:04:472022-11-22 09:12:522020-12-14 08:39:36enJournal articlesapplication/pdf1Let 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.