High powers of random elements of compact Lie groups


If a random unitary matrix \(U\) is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of \(U\) falling in a given arc, as the dimension of \(U\) tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9].

This is a preview of subscription content, access via your institution.

Author information



Additional information

Received: 15 October 1995 / In revised form: 7 March 1996

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rains, E. High powers of random elements of compact Lie groups. Probab Theory Relat Fields 107, 219–241 (1997). https://doi.org/10.1007/s004400050084

Download citation

  • Mathematics Subject Classification (1991):60B15, 22E99