Galois groups of random additive polynomials

We study the distribution of the Galois group of a random $q$-additive polynomial over a rational function field: For $q$ a power of a prime $p$, let $f=X^{q^n}+a_{n-1}{X}^{q^{n-1}}+\dots +a_1{X}^q+a_{0}X$ be a random polynomial chosen uniformly from the set of $q$-additive polynomials of degree $n$ and height $d$, that is, the coefficients are independent uniform polynomials of degree $\mathrm{d}\mathrm{e}\mathrm{g}{a}_i\le d$. The Galois group $G_f$ is a random subgroup of ${\mathrm{G}\mathrm{L}}_n(q)$. Our main result shows that $G_f$ is almost surely large as $d,q$ are fixed and $n\to \mathrm{\infty }$. For example, we give necessary and sufficient conditions so that ${\mathrm{S}\mathrm{L}}_n(q)\le G_f$ asymptotically almost surely. Our proof uses the classification of maximal subgroups of ${\mathrm{G}\mathrm{L}}_n(q)$. We also consider the limits: $q,n$ fixed, $d\to \mathrm{\infty }$ and $d,n$ fixed, $q\to \mathrm{\infty }$, which are more elementary.