Extreme eigenvalues of random matrices from Jacobi ensembles
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929218675277824 |
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| author | Winn, B. |
| author_facet | Winn, B. |
| contents | Two-term asymptotic formulae for the probability distribution functions for the smallest eigenvalue of the Jacobi $ β$-Ensembles are derived for matrices of large size in the régime where $ β> 0 $ is arbitrary and one of the model parameters $ α_1 $ is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases $ β= 2 $ and/or small values of $ α_1 $, explicit formulae involving more familiar functions, such as the modified Bessel function of the first kind, are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12082 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extreme eigenvalues of random matrices from Jacobi ensembles Winn, B. Probability 60B20 Two-term asymptotic formulae for the probability distribution functions for the smallest eigenvalue of the Jacobi $ β$-Ensembles are derived for matrices of large size in the régime where $ β> 0 $ is arbitrary and one of the model parameters $ α_1 $ is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases $ β= 2 $ and/or small values of $ α_1 $, explicit formulae involving more familiar functions, such as the modified Bessel function of the first kind, are presented. |
| title | Extreme eigenvalues of random matrices from Jacobi ensembles |
| topic | Probability 60B20 |
| url | https://arxiv.org/abs/2302.12082 |