Asymptotic freeness through unitaries generated by polynomials of Wigner matrices

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to N, the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in 9. More precisely given P a self-adjoint non-commutative polynomial and YN a d-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator P(YN) yields asymptotic freeness for large times.