Asymptotics of powers of random elements of compact Lie groups

2022/04/25 by Phillips, Donnelly
#60B20 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2204.11796

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≥ 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.

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