Asymptotics of powers of random elements of compact Lie groups

TL;DR

This paper extends Rains' results on the eigenvalue distributions of powers of Haar-distributed elements in compact Lie groups to more general distributions, identifying conditions for convergence and analyzing the limiting distribution of the powers.

Contribution

It introduces a mild absolute continuity condition ensuring eigenvalue distribution convergence for powers of general group elements and studies the limiting distribution of these powers.

Findings

Eigenvalue distribution of powers converges under the new condition.

Explicit limiting distributions are characterized.

Results generalize previous Haar measure findings.

Abstract

For a Haar-distributed element $H$ of a compact Lie group \(L\), Eric Rains proved that there is a natural number $D = D_L$ such that, for all $d\ge D$, the eigenvalue distribution of $H^d$ is fixed, and Rains described this fixed eigenvalue distribution explicitly. In the present paper we consider random elements $U$ of a compact Lie group with general distribution. In particular, we introduce a mild absolute continuity condition under which the eigenvalue distribution of powers of $U$ converges to that of $H^D$. Then, rather than the eigenvalue distribution, we consider the limiting distribution of $U^d$ itself.