Limiting spectral laws for sparse random circulant matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915250059608064 |
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| author | Beker, Adrian |
| author_facet | Beker, Adrian |
| contents | Fix a positive integer $d$ and let $(G_n)_{n\geq1}$ be a sequence of finite abelian groups with orders tending to infinity. For each $n \geq 1$, let $C_n$ be a uniformly random $G_n$-circulant matrix with entries in $\{0,1\}$ and exactly $d$ ones in each row/column. We show that the empirical spectral distribution of $C_n$ converges weakly in expectation to a probability measure $μ$ on $\mathbb{C}$ if and only if the distribution of the order of a uniform random element of $G_n$ converges weakly to a probability measure $ρ$ on $\mathbb{N}^*$, the one-point compactification of the natural numbers. Furthermore, we show that convergence in expectation can be strengthened to convergence in probability if and only if $ρ$ is a Dirac mass $δ_m$. In this case, $μ$ is the $d$-fold convolution of the uniform distribution on the $m$-th roots of unity if $m\in\mathbb{N}$ or the unit circle if $m = \infty$. We also establish that, under further natural assumptions, the determinant of $C_n$ is $\pm\exp((c_{m,d}+o(1))|G_n|)$ with high probability, where $c_{m,d}$ is a constant depending only on $m$ and $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limiting spectral laws for sparse random circulant matrices Beker, Adrian Probability Combinatorics Fix a positive integer $d$ and let $(G_n)_{n\geq1}$ be a sequence of finite abelian groups with orders tending to infinity. For each $n \geq 1$, let $C_n$ be a uniformly random $G_n$-circulant matrix with entries in $\{0,1\}$ and exactly $d$ ones in each row/column. We show that the empirical spectral distribution of $C_n$ converges weakly in expectation to a probability measure $μ$ on $\mathbb{C}$ if and only if the distribution of the order of a uniform random element of $G_n$ converges weakly to a probability measure $ρ$ on $\mathbb{N}^*$, the one-point compactification of the natural numbers. Furthermore, we show that convergence in expectation can be strengthened to convergence in probability if and only if $ρ$ is a Dirac mass $δ_m$. In this case, $μ$ is the $d$-fold convolution of the uniform distribution on the $m$-th roots of unity if $m\in\mathbb{N}$ or the unit circle if $m = \infty$. We also establish that, under further natural assumptions, the determinant of $C_n$ is $\pm\exp((c_{m,d}+o(1))|G_n|)$ with high probability, where $c_{m,d}$ is a constant depending only on $m$ and $d$. |
| title | Limiting spectral laws for sparse random circulant matrices |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2504.13833 |