Local laws and spectral properties of deformed sparse random matrices
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918473354969088 |
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| author | Lee, Ji Oon Yeo, Inyoung |
| author_facet | Lee, Ji Oon Yeo, Inyoung |
| contents | We consider deformed sparse random matrices of the form $H= W+ λV$, where $W$ is a real symmetric sparse random matrix, $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$, and $λ= O(1) $ is a coupling constant. Under mild assumptions on the matrix entries of $W$ and $V$, we prove local laws for $H$ that compare the empirical spectral measure of it with a refined version of the deformed semicircle law. By applying the local laws, we also prove several spectral properties of $H$, including the rigidity of the eigenvalues and the asymptotic normality of the extremal eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local laws and spectral properties of deformed sparse random matrices Lee, Ji Oon Yeo, Inyoung Probability 15B52, 60B20 We consider deformed sparse random matrices of the form $H= W+ λV$, where $W$ is a real symmetric sparse random matrix, $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$, and $λ= O(1) $ is a coupling constant. Under mild assumptions on the matrix entries of $W$ and $V$, we prove local laws for $H$ that compare the empirical spectral measure of it with a refined version of the deformed semicircle law. By applying the local laws, we also prove several spectral properties of $H$, including the rigidity of the eigenvalues and the asymptotic normality of the extremal eigenvalues. |
| title | Local laws and spectral properties of deformed sparse random matrices |
| topic | Probability 15B52, 60B20 |
| url | https://arxiv.org/abs/2507.02298 |