The circular law for sparse random combinatorial matrices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917401227952128 |
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| author | Li, Dongbin Litvak, Alexander E. Yu, Tingzhou |
| author_facet | Li, Dongbin Litvak, Alexander E. Yu, Tingzhou |
| contents | Let $\log^{2+\varepsilon} n \le d \le n/2$ for some fixed $\varepsilon \in (0,1)$, and let $M_n$ be an $n\times n$ random matrix with entries in ${0,1}$, where each row is independently and uniformly sampled from the set of all vectors in ${0,1}^n$ containing exactly $d$ ones. We show that the empirical spectral distribution of the appropriately rescaled matrix $M_n$ converges in probability to the circular law provided that $d=o(n)$. As a crucial element of the proof, we obtain quantitative lower bounds on the smallest singular value of the shifted matrices $M_n-zI_n$ whenever $|z|\le \sqrt d \log\log d$ and $C\log n \le d \le n/2$ for some absolute positive constant $C$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_10446 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The circular law for sparse random combinatorial matrices Li, Dongbin Litvak, Alexander E. Yu, Tingzhou Probability 60B20, 15B52, 60B10 (Primary), 60C05, 05C80, 46B06 (Secondary) Let $\log^{2+\varepsilon} n \le d \le n/2$ for some fixed $\varepsilon \in (0,1)$, and let $M_n$ be an $n\times n$ random matrix with entries in ${0,1}$, where each row is independently and uniformly sampled from the set of all vectors in ${0,1}^n$ containing exactly $d$ ones. We show that the empirical spectral distribution of the appropriately rescaled matrix $M_n$ converges in probability to the circular law provided that $d=o(n)$. As a crucial element of the proof, we obtain quantitative lower bounds on the smallest singular value of the shifted matrices $M_n-zI_n$ whenever $|z|\le \sqrt d \log\log d$ and $C\log n \le d \le n/2$ for some absolute positive constant $C$. |
| title | The circular law for sparse random combinatorial matrices |
| topic | Probability 60B20, 15B52, 60B10 (Primary), 60C05, 05C80, 46B06 (Secondary) |
| url | https://arxiv.org/abs/2604.10446 |