Cambridge Journal of Mathematics

Volume 10 (2022)

Number 1

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

Pages: 195 – 260



Benoît Collins (Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan)

Alice Guionnet (Université de Lyon, CNRS, ENSL, Lyon, France)

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


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.

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