The fluctuations of the mod p rank of triangular matrices

We consider random lower triangular matrices such that the entries on and below the diagonal are i.i.d. copies of some dollar sign double struck upper Z dollar sign $\mathbb{Z}$ $ ℤ $ -valued random variable. We prove that the Sylow dollar sign p dollar sign $p$ $ 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 dollar sign double struck upper F p dollar sign $\mathbb{F}_p$ $ 𝔽 p $ with i.i.d. random entries on and below the diagonal.