Quantitative gap universality for Wigner matrices
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911170829484032 |
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| author | Zhang, Albert |
| author_facet | Zhang, Albert |
| contents | We obtain the explicit rate of convergence $N^{-1/2 + ε}$ for the gaps of generalized Wigner matrices in the bulk of the spectrum, for distributions of matrix entries possibly atomic and supported on enough points. The proof proceeds by a Green function comparison, coupled with the relaxation estimate from [5]. In particular, we extend the 4 moment matching method [33] to arbitrary moments, allowing to compare resolvents down to the submicroscopic scale $N^{-3/2 + ε}$. This method also gives universality of the smallest gaps between eigenvalues for the Hermitian symmetry class, providing a universal, optimal separation of eigenvalues for discrete random matrices with entries supported on $Ω(1)$ points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative gap universality for Wigner matrices Zhang, Albert Probability Mathematical Physics We obtain the explicit rate of convergence $N^{-1/2 + ε}$ for the gaps of generalized Wigner matrices in the bulk of the spectrum, for distributions of matrix entries possibly atomic and supported on enough points. The proof proceeds by a Green function comparison, coupled with the relaxation estimate from [5]. In particular, we extend the 4 moment matching method [33] to arbitrary moments, allowing to compare resolvents down to the submicroscopic scale $N^{-3/2 + ε}$. This method also gives universality of the smallest gaps between eigenvalues for the Hermitian symmetry class, providing a universal, optimal separation of eigenvalues for discrete random matrices with entries supported on $Ω(1)$ points. |
| title | Quantitative gap universality for Wigner matrices |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2507.20442 |