The fluctuations of the mod p rank of triangular matrices

TL;DR

This paper studies the probabilistic fluctuations in the p-rank of random lower triangular matrices with integer entries, revealing their asymptotic behavior and connections to cokernel structures.

Contribution

It establishes the limiting fluctuation behavior of the p-rank of such matrices, extending previous results on matrix products and cokernels.

Findings

The Sylow p-subgroups of cokernels exhibit constant order fluctuations.

The p-rank fluctuations of lower triangular matrices over \\mathbb{F}_p are characterized asymptotically.

Connections are made between matrix cokernels and product fluctuation models.

Abstract

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 that of the matrix products studied by Nguyen and Van Peski. 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.