Galois groups of random polynomials over the rational function field

For a fixed prime power $q$ and natural number $d$ we consider a random polynomial $$f=x^n+a_{n-1}(t)x^{n-1}+\ldots+a_1(t)x+a_0(t)\in\mathbb F_q[t][x]$$ with $a_i$ drawn uniformly and independently at random from the set of all polynomials in $\mathbb F_q[t]$ of degree $\le d$. We show that with probability tending to 1 as $n\to\infty$ the Galois group $G_f$ of $f$ over $\mathbb F_q(t)$ is isomorphic to $S_{n-k}\times C$, where $C$ is cyclic, $k$ and $|C|$ are small quantities with a simple explicit dependence on $f$. As a corollary we deduce that $\mathbb P(G_f=S_n\,|\,f\mbox{ irreducible})\to 1$ as $n\to\infty$. Thus we are able to overcome the $S_n$ versus $A_n$ ambiguity in the most natural small box random polynomial model over $\mathbb F_q[t]$, which has not been achieved over $\mathbb Z$ so far.