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

Let XN = (X₁N,, XNd) be a d-tuple of N× N independent GUE random matrices and ZNM be any family of deterministic matrices in MN(C)⊗ MM(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(XN) 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/MNTr( f(P(XN⊗ IM,ZNM)) ) and its limit when N goes to infinity. If f is six times differentiable, we show that it is bounded by M² fC⁶N⁻². As a corollary we obtain a new proof of a result of Haagerup and Thorbj rnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in (XN,ZNM,ZNM^*) to converge almost surely towards its free limit. Restricting ourselves to polynomials in independent GUE matrices, we give concentration estimates on the largest eingenvalue of these polynomials around their free limit. A direct consequence of these inequalities is that there exists some β>0 such that for any ε₁<3+β)⁻¹ and ε₂<1/4, almost surely for N large enough, -1Nε₁\ ≤ \| P(XN)\| - P(x) ≤\ 1Nε₂. Finally if XN and YMN are independent and MN = o(N1/3), then almost surely, the norm of any polynomial in (XN⊗ IMN,IN⊗ YMN) converges almost surely towards its free limit. This result is an improvement of a Theorem of Pisier, who was himself using estimates from Haagerup and Thorbj rnsen, where MN had size o(N1/4).