The central limit theorem for entries of random matrices with specific rank over finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916396686901248 |
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| author | Chan, Chin Hei Xiong, Maosheng |
| author_facet | Chan, Chin Hei Xiong, Maosheng |
| contents | Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathcal{A}$ a non-empty proper subset of $\mathbb{F}_q$. Let $\mathbf{M}$ be a random $m \times n$ matrix of rank $r$ over $\mathbb{F}_q$ taken with uniform distribution. It was proved recently by Sanna that as $m,n \to \infty$ and $r,q,\mathcal{A}$ are fixed, the number of entries of $\mathbf{M}$ in $\mathcal{A}$ approaches a normal distribution. The question was raised as to whether or not one can still obtain a central limit theorem of some sort when $r$ goes to infinity in a way controlled by $m$ and $n$. In this paper we answer this question affirmatively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_10412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The central limit theorem for entries of random matrices with specific rank over finite fields Chan, Chin Hei Xiong, Maosheng Number Theory Combinatorics 15B52, 11T99, 05C50, 60F05 Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathcal{A}$ a non-empty proper subset of $\mathbb{F}_q$. Let $\mathbf{M}$ be a random $m \times n$ matrix of rank $r$ over $\mathbb{F}_q$ taken with uniform distribution. It was proved recently by Sanna that as $m,n \to \infty$ and $r,q,\mathcal{A}$ are fixed, the number of entries of $\mathbf{M}$ in $\mathcal{A}$ approaches a normal distribution. The question was raised as to whether or not one can still obtain a central limit theorem of some sort when $r$ goes to infinity in a way controlled by $m$ and $n$. In this paper we answer this question affirmatively. |
| title | The central limit theorem for entries of random matrices with specific rank over finite fields |
| topic | Number Theory Combinatorics 15B52, 11T99, 05C50, 60F05 |
| url | https://arxiv.org/abs/2409.10412 |