Asymptotics of powers of random elements of compact Lie groups
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866929233055449088 |
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| author | Phillips, Donnelly |
| author_facet | Phillips, Donnelly |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_11796 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotics of powers of random elements of compact Lie groups Phillips, Donnelly Probability 60B20 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. |
| title | Asymptotics of powers of random elements of compact Lie groups |
| topic | Probability 60B20 |
| url | https://arxiv.org/abs/2204.11796 |