Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913249199390720 |
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| author | Cipolloni, Giorgio Endős, László Schröder, Dominik |
| author_facet | Cipolloni, Giorgio Endős, László Schröder, Dominik |
| contents | We prove that the mesoscopic linear statistics $\sum_i f(n^a(σ_i-z_0))$ of the eigenvalues $\{σ_i\}_i$ of large $n\times n$ non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any $H^{2}_0$-functions $f$ around any point $z_0$ in the bulk of the spectrum on any mesoscopic scale $0<a<1/2$. This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, $a=0$, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of $X$ at spectral parameters $z_1, z_2$ with an improved error term in the entire mesoscopic regime $|z_1-z_2|\gg n^{-1/2}$. The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_12060 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices Cipolloni, Giorgio Endős, László Schröder, Dominik Probability Mathematical Physics We prove that the mesoscopic linear statistics $\sum_i f(n^a(σ_i-z_0))$ of the eigenvalues $\{σ_i\}_i$ of large $n\times n$ non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any $H^{2}_0$-functions $f$ around any point $z_0$ in the bulk of the spectrum on any mesoscopic scale $0<a<1/2$. This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, $a=0$, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of $X$ at spectral parameters $z_1, z_2$ with an improved error term in the entire mesoscopic regime $|z_1-z_2|\gg n^{-1/2}$. The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator. |
| title | Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2210.12060 |