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We consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices and we show that its $L^p$-norm can be upper bounded, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free semicircular variables as long as $p=o(N^{2/3})$.
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Parraud, F. (2022). On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices. Probability Theory and Related Fields , 182(3), 751-806
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Bandeira, A. S., Boedihardjo, M. T., & van Handel, R. (2023). Matrix concentration inequalities and free probability. Inventiones mathematicae , 234(1), 419-487
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Collins, B., Guionnet, A., & Parraud, F. (2022). On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices. Cambridge Journal of Mathematics , 10(1), 195–260
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Parraud, F. (2022). Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices. Communications in Mathematical Physics , 1-46
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Fronk, J., Krüger, T., & Nemish, Y. (2024). Norm Convergence Rate for Multivariate Quadratic Polynomials of Wigner Matrices. Journal of Functional Analysis , 110647
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