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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2411.12149 |
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| _version_ | 1866912963836772352 |
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| author | Keating, David Xu, Jiaming |
| author_facet | Keating, David Xu, Jiaming |
| contents | It is well known that the edge limit of Gaussian/Laguerre Beta-ensembles, as well as a large class of $β$-ensembles is given by the $\mathrm{Airy}(β)$ point process. We extend this universality result to a general class of additions of Gaussian and Laguerre ensembles, which were identified in \cite{AN} as projection of the ergodic measures of the $β$-corners process. In order to make sense of the $β$-addition, we introduce the Type-A Bessel function as the characteristic function of our matrix ensemble, following the approach of \cite{GM}, \cite{BCG}. Then we extract its moment information through the action of Dunkl operators, and obtain certain limiting functional expressed via conditional Brownian bridges for the Laplace transform of $\mathrm{Airy}(β)$. Our limit expression is universal up to proper rescaling among all of our additions, and agrees with the single-time Laplace transform expression from the concurrent work \cite{GXZ}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12149 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Airy limit for $β$-additions through Dunkl operators Keating, David Xu, Jiaming Probability Mathematical Physics Combinatorics It is well known that the edge limit of Gaussian/Laguerre Beta-ensembles, as well as a large class of $β$-ensembles is given by the $\mathrm{Airy}(β)$ point process. We extend this universality result to a general class of additions of Gaussian and Laguerre ensembles, which were identified in \cite{AN} as projection of the ergodic measures of the $β$-corners process. In order to make sense of the $β$-addition, we introduce the Type-A Bessel function as the characteristic function of our matrix ensemble, following the approach of \cite{GM}, \cite{BCG}. Then we extract its moment information through the action of Dunkl operators, and obtain certain limiting functional expressed via conditional Brownian bridges for the Laplace transform of $\mathrm{Airy}(β)$. Our limit expression is universal up to proper rescaling among all of our additions, and agrees with the single-time Laplace transform expression from the concurrent work \cite{GXZ}. |
| title | Airy limit for $β$-additions through Dunkl operators |
| topic | Probability Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2411.12149 |