Irreducible linear subgroups generated by pairs of matrices with large irreducible submodules

We call an element of a finite general linear group $\textrm{GL}(d,q)$ \emph{fat} if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than $d/2$. Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial has an irreducible factor of degree greater than $d/2$. We show that for groups $G$ with $\textrm{SL}(d,q) \leq G \leq \textrm{GL}(d,q)$ most pairs of fat elements from $G$ generate irreducible subgroups, namely we prove that the proportion of pairs of fat elements generating a reducible subgroup, in the set of all pairs in $G \times G$, is less than $q^{-d+1}$. We also prove that the conditional probability to obtain a pair $(g_1,g_2)$ in $G \times G$ which generates a reducible subgroup, given that $g_1, g_2$ are fat elements, is less than $2q^{-d+1}$. Further, we show that any reducible subgroup generated by a pair of fat elements acts irreducibly on a subspace of dimension greater than $d/2$, and in the induced action the generating pair corresponds to a pair of fat elements.