Proceedings of the american mathematical society vol:131 issue:7 pages:1981-1988
Let H-0 be an arbitrary self-adjoint n x n matrix and H( n) be an n x n (random) Wigner matrix. We show that t bar right arrow Tr exp(H (n) - i t H-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 bar right arrow tau (exp(a - i tb)) is positive definite whenever the noncommutative random variables a and b are in free relation, with a semicircular.