Recovering Parameters from Edge Fluctuations: Beta-Ensembles and Critically-Spiked Models
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914189772062720 |
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| author | Lamarre, Pierre Yves Gaudreau |
| author_facet | Lamarre, Pierre Yves Gaudreau |
| contents | Let $Λ=\{Λ_0,Λ_1,Λ_2,\ldots\}$ be the point process that describes the edge scaling limit of either (i) "regular" beta-ensembles with inverse temperature $β>0$, or (ii) the top eigenvalues of Wishart or Gaussian invariant random matrices perturbed by $r_0\geq1$ critical spikes. In other words, $Λ$ is the eigenvalue point process of one of the scalar or multivariate stochastic Airy operators. We prove that a single observation of $Λ$ suffices to recover (almost surely) either (i) $β$ in the case of beta-ensembles, or (ii) $r_0$ in the case of critically-spiked models. Our proof relies on the recently-developed semigroup theory for the multivariate stochastic Airy operators.
Going beyond these parameter-recovery applications, our results also (iii) refine our understanding of the rigidity properties of $Λ$, and (iv) shed new light on the equality (in distribution) of stochastic Airy spectra with different dimensions and the same Robin boundary conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recovering Parameters from Edge Fluctuations: Beta-Ensembles and Critically-Spiked Models Lamarre, Pierre Yves Gaudreau Probability Mathematical Physics 60H25, 47D08, 60B20, 62E20, 60K35 Let $Λ=\{Λ_0,Λ_1,Λ_2,\ldots\}$ be the point process that describes the edge scaling limit of either (i) "regular" beta-ensembles with inverse temperature $β>0$, or (ii) the top eigenvalues of Wishart or Gaussian invariant random matrices perturbed by $r_0\geq1$ critical spikes. In other words, $Λ$ is the eigenvalue point process of one of the scalar or multivariate stochastic Airy operators. We prove that a single observation of $Λ$ suffices to recover (almost surely) either (i) $β$ in the case of beta-ensembles, or (ii) $r_0$ in the case of critically-spiked models. Our proof relies on the recently-developed semigroup theory for the multivariate stochastic Airy operators. Going beyond these parameter-recovery applications, our results also (iii) refine our understanding of the rigidity properties of $Λ$, and (iv) shed new light on the equality (in distribution) of stochastic Airy spectra with different dimensions and the same Robin boundary conditions. |
| title | Recovering Parameters from Edge Fluctuations: Beta-Ensembles and Critically-Spiked Models |
| topic | Probability Mathematical Physics 60H25, 47D08, 60B20, 62E20, 60K35 |
| url | https://arxiv.org/abs/2503.14414 |