Law of fractional logarithm for random matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914068805189632 |
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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 prove the Paquette-Zeitouni law of fractional logarithm (LFL) for the extreme eigenvalues [arXiv:1505.05627] in full generality, and thereby verify a conjecture from [arXiv:1505.05627]. Our result holds for any Wigner minor process and both symmetry classes, in particular for the GOE minor process, while [arXiv:1505.05627] and the recent full resolution of LFL by Baslingker et.~al.~[arXiv:2410.11836] cover only the GUE case which is determinantal. Lacking the possibility for a direct comparison with the Gaussian case, we develop a robust and natural method for both key parts of the proof. On one hand, we rely on a powerful martingale technique to describe precisely the strong correlation between the largest eigenvalue of an $N\times N$ Wigner matrix and its $(N-k)\times (N-k)$ minor if $k\ll N^{2/3}$. On the other hand, we use dynamical methods to show that this correlation is weak if $k\gg N^{2/3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Law of fractional logarithm for random matrices Bao, Zhigang Cipolloni, Giorgio Erdős, László Henheik, Joscha Kolupaiev, Oleksii Probability Mathematical Physics 60B20, 60G55, 82C10 We prove the Paquette-Zeitouni law of fractional logarithm (LFL) for the extreme eigenvalues [arXiv:1505.05627] in full generality, and thereby verify a conjecture from [arXiv:1505.05627]. Our result holds for any Wigner minor process and both symmetry classes, in particular for the GOE minor process, while [arXiv:1505.05627] and the recent full resolution of LFL by Baslingker et.~al.~[arXiv:2410.11836] cover only the GUE case which is determinantal. Lacking the possibility for a direct comparison with the Gaussian case, we develop a robust and natural method for both key parts of the proof. On one hand, we rely on a powerful martingale technique to describe precisely the strong correlation between the largest eigenvalue of an $N\times N$ Wigner matrix and its $(N-k)\times (N-k)$ minor if $k\ll N^{2/3}$. On the other hand, we use dynamical methods to show that this correlation is weak if $k\gg N^{2/3}$. |
| title | Law of fractional logarithm for random matrices |
| topic | Probability Mathematical Physics 60B20, 60G55, 82C10 |
| url | https://arxiv.org/abs/2503.18922 |