Perturbation of Wigner matrices and a conjecture

Let $H_{0}$ be an arbitrary self-adjoint $n\times n$ matrix and $H(n)$ be an $n\times n$ (random) Wigner matrix. We show that $t\mapsto \hbox {Tr}\, \exp (H(n)-\text{i}\, tH_{0})$ is positive definite in the average. This partially answers a long-standing conjecture. On the basis of asymptotic freeness our result implies that $t\mapsto \tau (\exp (a- \text{i}\, tb))$ is positive definite whenever the noncommutative random variables $a$ and $b$ are in free relation, with $a$ semicircular.