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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.30779 |
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| _version_ | 1866917546994696192 |
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| author | Bao, Zhigang Wang, Dong Zhu, Yue |
| author_facet | Bao, Zhigang Wang, Dong Zhu, Yue |
| contents | We investigate the eigenvector distribution at the soft edge for Gaussian random matrices with finite-rank deformations, in the critical regime of BBP transition. For finite-rank deformations of the GOE and GUE with critical spikes, we find that the squared overlap between a leading eigenvector and a spike, rescaled by \(N^{1/3}\), converges weakly to the negative reciprocal of the derivative of an Airy-Green function evaluated at the corresponding soft-edge root. For the rank-one critically spiked Gaussian \(β\)-ensemble, \(β>0\), we obtain an analogous result involving an Airy-Green function. In both cases, the Airy-Green functions are generalizations of the one introduced by Bykhovskaya--Gorin--Sodin \cite{Bykhovskaya-Gorin-Sodin25}. The proofs are both based on an eigenvector--eigenvalue identity and a resolvent-differentiation mechanism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30779 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eigenvector distribution of random matrices under critical finite-rank deformations Bao, Zhigang Wang, Dong Zhu, Yue Probability We investigate the eigenvector distribution at the soft edge for Gaussian random matrices with finite-rank deformations, in the critical regime of BBP transition. For finite-rank deformations of the GOE and GUE with critical spikes, we find that the squared overlap between a leading eigenvector and a spike, rescaled by \(N^{1/3}\), converges weakly to the negative reciprocal of the derivative of an Airy-Green function evaluated at the corresponding soft-edge root. For the rank-one critically spiked Gaussian \(β\)-ensemble, \(β>0\), we obtain an analogous result involving an Airy-Green function. In both cases, the Airy-Green functions are generalizations of the one introduced by Bykhovskaya--Gorin--Sodin \cite{Bykhovskaya-Gorin-Sodin25}. The proofs are both based on an eigenvector--eigenvalue identity and a resolvent-differentiation mechanism. |
| title | Eigenvector distribution of random matrices under critical finite-rank deformations |
| topic | Probability |
| url | https://arxiv.org/abs/2605.30779 |