The smallest singular value of sparse discrete random matrices
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909702613368832 |
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| author | Yu, Kexin |
| author_facet | Yu, Kexin |
| contents | Let $M_n$ be an $n \times n$ random matrix with i.i.d. sparse discrete entries. In this paper, we develop a simple framework to solve the approximate Spielman-Teng theorem for $M_n$, which has the following form: There exist constants $C, c>0$ such that for all $η\geq 0$, $\textbf{P}\left(s_n\left(M_n\right) \leq η\right) \lesssim n^C η+\exp \left(-n^c\right)$. As an application, we give an approximate Spielman-Teng theorem for $M_n$ whose entries are $μ-$lazy random variables, extending previous work by Tao and Vu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17741 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The smallest singular value of sparse discrete random matrices Yu, Kexin Probability Let $M_n$ be an $n \times n$ random matrix with i.i.d. sparse discrete entries. In this paper, we develop a simple framework to solve the approximate Spielman-Teng theorem for $M_n$, which has the following form: There exist constants $C, c>0$ such that for all $η\geq 0$, $\textbf{P}\left(s_n\left(M_n\right) \leq η\right) \lesssim n^C η+\exp \left(-n^c\right)$. As an application, we give an approximate Spielman-Teng theorem for $M_n$ whose entries are $μ-$lazy random variables, extending previous work by Tao and Vu. |
| title | The smallest singular value of sparse discrete random matrices |
| topic | Probability |
| url | https://arxiv.org/abs/2507.17741 |