Asymptotic expansion of smooth functions in deterministic and iid Haar unitary matrices, and application to tensor products of matrices

Let UN be a family of N×N independent Haar unitary random matrices and their adjoints, ZN a family of deterministic matrices, and P a self-adjoint noncommutative polynomial, that is, for any N, P(UN,ZN) is self-adjoint, f a smooth function. We prove that for any k, if f is smooth enough, there exist deterministic constants αiP(f,ZN) such that E[1 NTr(f(P(UN,ZN)))]=∑i=0kαiP(f,ZN)N2i+O(N−2k−2). Besides, the constants αiP(f,ZN) are built explicitly with the help of free probability. As a corollary, we prove that given α<1/2, for N large enough, every eigenvalue of P(UN,ZN) is N−α-close to the spectrum of P(u,ZN) where u is a d-tuple of free Haar unitaries. We also prove the convergence of the norm of any polynomial P(UN⊗IM,IN⊗YM) as long as the family YM converges strongly and that M≪Nln−3(N).