Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field

Abstract

We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform model studied by Fulman in his thesis. The sharp sparsity threshold for the full rational canonical form coincides with the threshold previously obtained by Lee for finite-field cokernels and by Jung-Lee-Yu for random matrix models over $\mathbb{Z}_p$. Our proof is based on the surjection moment method over the function field $\mathbb{F}_p[t]$, applied to the finite module $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$, whose primary decomposition records the rational canonical form. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold $1/d$ for statistics associated with irreducible polynomials of minimal degree $d$.