Quenched universality for deformed Wigner matrices
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929315944333312 |
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| author | Cipolloni, Giorgio Erdős, László Schröder, Dominik |
| author_facet | Cipolloni, Giorgio Erdős, László Schröder, Dominik |
| contents | Following E. Wigner's original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix $H$ yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices $H+xA$ with a deterministic Hermitian matrix $A$ and a fixed Wigner matrix $H$, just using the randomness of a single scalar real random variable $x$. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_10200 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quenched universality for deformed Wigner matrices Cipolloni, Giorgio Erdős, László Schröder, Dominik Mathematical Physics Probability 60B20, 15B52 Following E. Wigner's original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix $H$ yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices $H+xA$ with a deterministic Hermitian matrix $A$ and a fixed Wigner matrix $H$, just using the randomness of a single scalar real random variable $x$. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble. |
| title | Quenched universality for deformed Wigner matrices |
| topic | Mathematical Physics Probability 60B20, 15B52 |
| url | https://arxiv.org/abs/2106.10200 |