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

Félix Parraud (Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan; and Université de Lyon, ENSL, UMPA, Lyon, France)

Abstract

Let $X^N = (X^N_1 , \dotsc , X^N_d)$ be a d‑tuple of $N \times N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N \mathbb{C}) \otimes \mathbb{M}_M (\mathbb{C})$. 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\[\frac{1}{MN} \mathrm{Tr}_N \otimes \mathrm{Tr}_M \bigl( f(P(X^N \otimes I_M, Z^{NM})) \bigr) \: ,\]and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2 {\lVert f \rVert}_{\mathcal{C}^6} N^{-2}$. 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 $(X^N, Z^{NM}, Z^{NM^\ast})$ to converge almost surely towards its free limit.

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B. C. was partially funded by JSPS KAKENHI 17K18734, 17H04823, 15KK0162. F. P. also benefited from the same Kakenhi grants and a MEXT JASSO fellowship. A. G. and F. P. were partially supported by Labex Milyon (ANR-10-LABX-0070) of Université de Lyon and the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 884584.

Received 6 March 2020

Published 21 April 2022