Universality of rational canonical form for random matrices over a finite field

In this note, we study the distribution of the rational canonical form of a random matrix over the finite field ${\mathbb{F}}_p$, whose entries are independent and $\epsilon$-balanced with $\epsilon \in (0,1-1\mathrm{/}p]$. We show that, as the matrix size tends to infinity, the statistics converge to independent Cohen-Lenstra distributions, demonstrating the universality of this asymptotic behavior. In particular, we recover, as a special case, the uniform setting proved by Fulman in his thesis in 1997. Our proof uses the fact that the rational canonical form data of $A_n$ and the ${\mathbb{F}}_p[t]$-module structure of the function field cokernel $\text{Cok}(t{I}_n-A_n)$ determine each other uniquely. Consequently, our question can be reformulated, equivalently, as the asymptotic distribution problem for this cokernel, which has been established by Cheong-Yu (arXiv:2303.09125).