Large $q$ convergence of random characteristic polynomials to random permutations and its applications.

We extend an observation due to Stong that the distribution of the number of degree $d$ irreducible factors of the characteristic polynomial of a random $n \times n$ matrix over a finite field $\mathbb{F}_{q}$ converges to the distribution of the number of length $d$ cycles of a random permutation in $S_{n}$, as $q \rightarrow \infty$, by having any finitely many choices of $d$, say $d_{1}, \dots, d_{r}$. This generalized convergence will be used for the following two applications: the distribution of the cokernel of an $n \times n$ Haar-random $\mathbb{Z}_{p}$-matrix when $p \rightarrow \infty$ and a matrix version of Landau's theorem that estimates the number of irreducible factors of a random characteristic polynomial for large $n$ when $q \rightarrow \infty$.