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

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 $α_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{nα_n}{\log n}>1, $$ then, for every fixed collection of distinct monic