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

Let X-N = (X-1(N) ,..., X-d(N)) be a d-tuple of N x N independent GUE random matrices and Z(NM) be any family of deterministic matrices in M-N(C) circle times 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-N circle times Tr-M (f(P(X-N circle times I-M, Z(NM)))), and its limit when N goes to infinity. If f is six times differentiable, we show that it is bounded by M-2 parallel to f parallel to(C6) N-2. As a corollary, we obtain a new proof and slightly improve a result of Haagerup and Thorbjornsen, 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.