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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.12082 |
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Table of 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.