Airy$_β$ line ensemble and its Laplace transform
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929594739720192 |
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| author | Gorin, Vadim Xu, Jiaming Zhang, Lingfu |
| author_facet | Gorin, Vadim Xu, Jiaming Zhang, Lingfu |
| contents | The Airy$_β$ line ensemble is a random collection of continuous curves, which should serve as a universal edge scaling limit in problems related to eigenvalues of random matrices and models of 2d statistical mechanics. This line ensemble unifies many existing universal objects including Tracy-Widom distributions, eigenvalues of the Stochastic Airy Operator, Airy$_2$ process from the KPZ theory. Here $β>0$ is a real parameter governing the strength of the repulsion between the curves.
We introduce and characterize the Airy$_β$ line ensemble in terms of the Laplace transform, by producing integral formulas for its joint multi-time moments. We prove two asymptotic theorems for each $β>0$: the trajectories of the largest eigenvalues in the Dyson Brownian Motion converge to the Airy$_β$ line ensemble; the extreme particles in the G$β$E corners process converge to the same limit.
The proofs are based on the convergence of random walk expansions for the multi-time moments of prelimit objects towards their Brownian counterparts. The expansions are produced through Dunkl differential-difference operators acting on multivariate Bessel generating functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10829 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Airy$_β$ line ensemble and its Laplace transform Gorin, Vadim Xu, Jiaming Zhang, Lingfu Probability Mathematical Physics The Airy$_β$ line ensemble is a random collection of continuous curves, which should serve as a universal edge scaling limit in problems related to eigenvalues of random matrices and models of 2d statistical mechanics. This line ensemble unifies many existing universal objects including Tracy-Widom distributions, eigenvalues of the Stochastic Airy Operator, Airy$_2$ process from the KPZ theory. Here $β>0$ is a real parameter governing the strength of the repulsion between the curves. We introduce and characterize the Airy$_β$ line ensemble in terms of the Laplace transform, by producing integral formulas for its joint multi-time moments. We prove two asymptotic theorems for each $β>0$: the trajectories of the largest eigenvalues in the Dyson Brownian Motion converge to the Airy$_β$ line ensemble; the extreme particles in the G$β$E corners process converge to the same limit. The proofs are based on the convergence of random walk expansions for the multi-time moments of prelimit objects towards their Brownian counterparts. The expansions are produced through Dunkl differential-difference operators acting on multivariate Bessel generating functions. |
| title | Airy$_β$ line ensemble and its Laplace transform |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2411.10829 |