Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices

Let $$X^N$$ be a family of $$N\times N$$ independent GUE random matrices, $$Z^N$$ deterministic P self-adjoint noncommutative polynomial, that is for any N, $$P(X^N,Z^N)$$ self-adjoint, f smooth function. We prove k, if enough, there exist constants $$\alpha _i^P(f,Z^N)$$ such $$\begin{aligned} \mathbb {E}\left[ \frac{1}{N}\text {Tr}\left( f(P(X^N,Z^N)) \right) \right] \ =\ \sum _{i=0}^k \frac{\alpha _i^P(f,Z^N)}{N^{2i}}\ +\ \mathcal {O}(N^{-2k-2}) . \end{aligned}$$ Besides, the are built explicitly with help free probability. In particular, x semicircular system, then when support and spectrum $$P(x,Z^N)$$ disjoint, _i^P(f,Z^N)=0$$ all $$i\in {N}$$ As corollary, we given <1/2$$ , N large every eigenvalue $$N^{-\alpha }$$ -close to