On the spectral radius and the characteristic polynomial of a random matrix with independent elements and a variance profile
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| Format: | Preprint |
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2025
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| _version_ | 1866911199026741248 |
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| author | Hachem, Walid Louvaris, Michail |
| author_facet | Hachem, Walid Louvaris, Michail |
| contents | In this paper, it is shown that with large probability, the spectral radius of a large non-Hermitian random matrix with a general variance profile does not exceed the square root of the spectral radius of the variance profile matrix. A minimal moment assumption is considered and sparse variance profiles are covered. Following an approach developed recently by Bordenave, Chafa{ï} and Garc{í}a-Zelada, the key theorem states the asymptotic equivalence between the reverse characteristic polynomial of the random matrix at hand and a random analytic function which depends on the variance profile matrix. The result is applied to the case of a non-Hermitian random matrix with a variance profile given by a piecewise constant or a continuous non-negative function, the inhomogeneous (centered) directed Erdős-R{é}nyi model, and more. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03657 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the spectral radius and the characteristic polynomial of a random matrix with independent elements and a variance profile Hachem, Walid Louvaris, Michail Probability In this paper, it is shown that with large probability, the spectral radius of a large non-Hermitian random matrix with a general variance profile does not exceed the square root of the spectral radius of the variance profile matrix. A minimal moment assumption is considered and sparse variance profiles are covered. Following an approach developed recently by Bordenave, Chafa{ï} and Garc{í}a-Zelada, the key theorem states the asymptotic equivalence between the reverse characteristic polynomial of the random matrix at hand and a random analytic function which depends on the variance profile matrix. The result is applied to the case of a non-Hermitian random matrix with a variance profile given by a piecewise constant or a continuous non-negative function, the inhomogeneous (centered) directed Erdős-R{é}nyi model, and more. |
| title | On the spectral radius and the characteristic polynomial of a random matrix with independent elements and a variance profile |
| topic | Probability |
| url | https://arxiv.org/abs/2501.03657 |