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 $Y^N$ a $d$-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator $P(Y^N)$ yields asymptotic freeness for large times.