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The fluctuations of the mod p rank of triangular matrices
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 19 May 2026, pp. 1-19
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- Article
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We consider random lower triangular matrices such that the entries on and below the diagonal are i.i.d. copies of some
$\mathbb{Z}$-valued random variable. We prove that the Sylow 
$p$-subgroups of the cokernels of these matrices have the same constant order fluctuations as those of the matrix products studied by Nguyen and Van Peski. Unlike for matrix products, for triangular matrices, the law of the limiting fluctuations depends slightly on the distribution of the entries. As a special case, we can describe the limiting fluctuations of the rank of lower triangular matrices over 
$\mathbb{F}_p$ with i.i.d. random entries on and below the diagonal.
