The smallest singular value of signed random combinatorial matrices
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910125857439744 |
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| author | Yu, Kexin |
| author_facet | Yu, Kexin |
| contents | Let $M_n$ be an $n\times n$ signed random combinatorial matrix whose rows are independent and uniformly distributed over the set of $\{-1,0,1\}$-vectors with exactly $n/2$ zero coordinates. Despite the dependence induced by the row constraints, we prove that there exist constants $C,c > 0$ such that for any $\varepsilon\ge0$, \begin{align*} \textbf{P}\left(s_{n}(M_n)\le {\varepsilon}{n^{-1/2}}\right)\le C\varepsilon+e^{-cn}. \end{align*} In particular, the probability that $M_n$ is singular is exponentially small. Our approach builds on the Combinatorial Least Common Denominator (CLCD) introduced by Tran and develops the method in the present constrained setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11761 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The smallest singular value of signed random combinatorial matrices Yu, Kexin Probability Combinatorics Let $M_n$ be an $n\times n$ signed random combinatorial matrix whose rows are independent and uniformly distributed over the set of $\{-1,0,1\}$-vectors with exactly $n/2$ zero coordinates. Despite the dependence induced by the row constraints, we prove that there exist constants $C,c > 0$ such that for any $\varepsilon\ge0$, \begin{align*} \textbf{P}\left(s_{n}(M_n)\le {\varepsilon}{n^{-1/2}}\right)\le C\varepsilon+e^{-cn}. \end{align*} In particular, the probability that $M_n$ is singular is exponentially small. Our approach builds on the Combinatorial Least Common Denominator (CLCD) introduced by Tran and develops the method in the present constrained setting. |
| title | The smallest singular value of signed random combinatorial matrices |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2604.11761 |