Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914472765947904 |
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| author | Cook, Nicholas A. Ducatez, Raphael Guionnet, Alice |
| author_facet | Cook, Nicholas A. Ducatez, Raphael Guionnet, Alice |
| contents | We establish precise upper-tail asymptotics and large deviation principles for the rightmost eigenvalue $λ_1$ of Wigner matrices with sub-Gaussian entries. In contrast to the case of heavier tails, where deviations of $λ_1$ are due to the appearance of a few large entries, and the sharp sub-Gaussian case that is governed by the collective deviation of entries in a delocalized rank-one pattern, we show that the general sub-Gaussian case is determined by a mixture of localized and delocalized effects.
Our key result is a finite-$N$ approximation for the upper tail of $λ_1$ by an optimization problem involving \emph{restricted annealed free energies} for a spherical spin glass model. This new type of argument allows us to derive full large deviation principles when the log-Laplace transform of the entries' distribution $μ$ has bounded second derivative, whereas previous results required much more restrictive assumptions, namely sharp sub-Gaussianity and symmetry, or only covered certain ranges of deviations.
We show that the sharp sub-Gaussian condition characterizes measures $μ$ for which the rate function coincides with that of the Gaussian Orthogonal Ensemble (GOE). When $μ$ is not sharp sub-Gaussian, at a certain distance from the bulk of the spectrum there is a transition from the GOE rate function to a non-universal rate function depending on $μ$, and this transition coincides with the onset of a localization phenomenon for the associated eigenvector. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_14823 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices Cook, Nicholas A. Ducatez, Raphael Guionnet, Alice Probability 60B20, 60F10 We establish precise upper-tail asymptotics and large deviation principles for the rightmost eigenvalue $λ_1$ of Wigner matrices with sub-Gaussian entries. In contrast to the case of heavier tails, where deviations of $λ_1$ are due to the appearance of a few large entries, and the sharp sub-Gaussian case that is governed by the collective deviation of entries in a delocalized rank-one pattern, we show that the general sub-Gaussian case is determined by a mixture of localized and delocalized effects. Our key result is a finite-$N$ approximation for the upper tail of $λ_1$ by an optimization problem involving \emph{restricted annealed free energies} for a spherical spin glass model. This new type of argument allows us to derive full large deviation principles when the log-Laplace transform of the entries' distribution $μ$ has bounded second derivative, whereas previous results required much more restrictive assumptions, namely sharp sub-Gaussianity and symmetry, or only covered certain ranges of deviations. We show that the sharp sub-Gaussian condition characterizes measures $μ$ for which the rate function coincides with that of the Gaussian Orthogonal Ensemble (GOE). When $μ$ is not sharp sub-Gaussian, at a certain distance from the bulk of the spectrum there is a transition from the GOE rate function to a non-universal rate function depending on $μ$, and this transition coincides with the onset of a localization phenomenon for the associated eigenvector. |
| title | Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices |
| topic | Probability 60B20, 60F10 |
| url | https://arxiv.org/abs/2302.14823 |