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Asymptotics of powers of random elements of compact Lie groups

Donnelly Phillips

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.

Asymptotics of powers of random elements of compact Lie groups

Abstract

For a Haar-distributed element of a compact Lie group , Eric Rains proved that there is a natural number such that, for all , the eigenvalue distribution of is fixed, and Rains described this fixed eigenvalue distribution explicitly. In the present paper we consider random elements 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 converges to that of . Then, rather than the eigenvalue distribution, we consider the limiting distribution of itself.
Paper Structure (9 sections, 16 theorems, 33 equations)

This paper contains 9 sections, 16 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Statement of main results
  4. Limiting distributions of eigenvalues
  5. Fourier analysis approach
  6. Standard measure theory approach
  7. Eigenvalue distributions
  8. Limiting distributions of powers of group elements
  9. Regarding random preimages

Key Result

Theorem 1.1

Let $L$ be a compact Lie group that has a maximal torus $T$ of dimension $n$, and let $\phi : L \rightarrow GL(V)$ be a unitary representation of $L$ on a vector space $V$ of dimension $N$. Let $C \subseteq L$ be a connected component of $L$, and suppose that $H$ is Haar-distributed on $C$. Let $\{Y

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark
  • Lemma 2.2
  • proof
  • Remark
  • proof : Proof of Theorem \ref{['torus1']}.
  • Lemma 2.3
  • ...and 28 more