No abstract

Let X N = (X N 1 , . . . , X N d ) be a d-tuple of N × N independent GUE random matrices and Z NM be any family of deterministic matrices in M N (C) ⊗ 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 ofand its limit when N goes to infinity. If f is six times differentiable, we show that it is bounded by M 2 f 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 * ) to converge almost surely towards its free limit.