Decorrelation transition in the Wigner minor process
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908539126022144 |
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| author | Bao, Zhigang Cipolloni, Giorgio Erdős, László Henheik, Joscha Kolupaiev, Oleksii |
| author_facet | Bao, Zhigang Cipolloni, Giorgio Erdős, László Henheik, Joscha Kolupaiev, Oleksii |
| contents | We consider the Wigner minor process, i.e. the eigenvalues of an $N\times N$ Wigner matrix $H^{(N)}$ together with the eigenvalues of all its $n\times n$ minors, $H^{(n)}$, $n\le N$. The top eigenvalues of $H^{(N)}$ and those of its immediate minor $H^{(N-1)}$ are very strongly correlated, but this correlation becomes weaker for smaller minors $H^{(N-k)}$ as $k$ increases. For the GUE minor process the critical transition regime around $k\sim N^{2/3}$ was analyzed by Forrester and Nagao (J. Stat. Mech.: Theory and Experiment, 2011) providing an explicit formula for the nontrivial joint correlation function. We prove that this formula is universal, i.e. it holds for the Wigner minor process. Moreover, we give a complete analysis of the sub- and supercritical regimes both for eigenvalues and for the corresponding eigenvector overlaps, thus we prove the decorrelation transition in full generality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_06549 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decorrelation transition in the Wigner minor process Bao, Zhigang Cipolloni, Giorgio Erdős, László Henheik, Joscha Kolupaiev, Oleksii Probability Mathematical Physics 60B20, 60G55, 82C10 We consider the Wigner minor process, i.e. the eigenvalues of an $N\times N$ Wigner matrix $H^{(N)}$ together with the eigenvalues of all its $n\times n$ minors, $H^{(n)}$, $n\le N$. The top eigenvalues of $H^{(N)}$ and those of its immediate minor $H^{(N-1)}$ are very strongly correlated, but this correlation becomes weaker for smaller minors $H^{(N-k)}$ as $k$ increases. For the GUE minor process the critical transition regime around $k\sim N^{2/3}$ was analyzed by Forrester and Nagao (J. Stat. Mech.: Theory and Experiment, 2011) providing an explicit formula for the nontrivial joint correlation function. We prove that this formula is universal, i.e. it holds for the Wigner minor process. Moreover, we give a complete analysis of the sub- and supercritical regimes both for eigenvalues and for the corresponding eigenvector overlaps, thus we prove the decorrelation transition in full generality. |
| title | Decorrelation transition in the Wigner minor process |
| topic | Probability Mathematical Physics 60B20, 60G55, 82C10 |
| url | https://arxiv.org/abs/2503.06549 |