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

Let $U^N = (U_1^N,\dots, U^N_p)$ be a d-tuple of $N\times N$ independent Haar unitary 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. In 1998, Voiculescu showed that the empirical measure of the eigenvalues of this polynomial evaluated in Haar unitary matrices and deterministic matrices 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 $$\frac{1}{MN} \text{Tr}\left( f(P(U^N\otimes I_M,Z^{NM})) \right) ,$$ and its limit when $N$ goes to infinity. If $f$ is seven times differentiable, we show that it is bounded by $M^2 \left\Vert f\right\Vert_{\mathcal{C}^7} N^{-2}$. As a corollary we obtain a new proof with quantitative bounds of a result of Collins and Male which gives sufficient conditions for the operator norm of a polynomial evaluated in Haar unitary matrices and deterministic matrices to converge almost surely towards its free limit. Actually we show that if $U^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(U^N\otimes I_{M_N}, I_N\otimes Y^{M_N})$ converges almost surely towards its free limit.