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

Abstract

We study the distribution of rational canonical form of a random matrix over the finite field $\mathbb{F}_p$, whose entries are independent and $ε$-balanced with $ε\in(0,1-1/p]$. 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. Our method builds on a function field version of Wood's surjection moment method (arXiv:1504.04391), and in particular it recovers, as a special case, the uniform model proved earlier by Fulman in his 1997 thesis.