The log-Characteristic Polynomial of Generalized Wigner Matrices is Log-Correlated
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914657786134528 |
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| author | Mody, Krishnan |
| author_facet | Mody, Krishnan |
| contents | We prove that in the limit of large dimension, the distribution of the logarithm of the characteristic polynomial of a generalized Wigner matrix converges to a log-correlated field. In particular, this shows that the limiting joint fluctuations of the eigenvalues are also log-correlated. Our argument mirrors that of \cite{BouMod2019}, which is in turn based on the three-step argument of \cite{ErdPecRmSchYau2010,ErdSchYau2011Uni}, but applies to a wider class of models, and at the edge of the spectrum. We rely on (i) the results in the Gaussian cases, special cases of the results in \cite{BouModPai2021}, (ii) the local laws of \cite{ErdYauYin2012}(iii) the observable \cite{Bou2020} introduced and its analysis of the stochastic advection equation this observable satisfies, and (iv) the argument for a central limit theorem on mesoscopic scales in \cite{LanLopSos2021}. For the proof, we also establish a Wegner estimate and local law down to the microscopic scale, both at the edge of the spectrum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_05956 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The log-Characteristic Polynomial of Generalized Wigner Matrices is Log-Correlated Mody, Krishnan Probability 15B52 We prove that in the limit of large dimension, the distribution of the logarithm of the characteristic polynomial of a generalized Wigner matrix converges to a log-correlated field. In particular, this shows that the limiting joint fluctuations of the eigenvalues are also log-correlated. Our argument mirrors that of \cite{BouMod2019}, which is in turn based on the three-step argument of \cite{ErdPecRmSchYau2010,ErdSchYau2011Uni}, but applies to a wider class of models, and at the edge of the spectrum. We rely on (i) the results in the Gaussian cases, special cases of the results in \cite{BouModPai2021}, (ii) the local laws of \cite{ErdYauYin2012}(iii) the observable \cite{Bou2020} introduced and its analysis of the stochastic advection equation this observable satisfies, and (iv) the argument for a central limit theorem on mesoscopic scales in \cite{LanLopSos2021}. For the proof, we also establish a Wegner estimate and local law down to the microscopic scale, both at the edge of the spectrum. |
| title | The log-Characteristic Polynomial of Generalized Wigner Matrices is Log-Correlated |
| topic | Probability 15B52 |
| url | https://arxiv.org/abs/2302.05956 |