HIGH POWERS OF RANDOM ELEMENTS OF COMPACT LIE-GROUPS

If a random unitary matrix U is raised to a sufficiently high power, i ts eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the v ariance of the number of eigenvalues of U falling in a given are, as t he dimension of U tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie gr oups. We state and prove this more general theorem, paying special att ention to the compact classical groups and to wreath products. This pa per is excerpted from the author's doctoral thesis, [9].