The proportion of non-degenerate complementary subspaces in classical spaces

Given positive integers $e_1,e_2$, let $X_i$ denote the set of $e_i$-dimensional subspaces of a fixed finite vector space $V=({\mathbb{F}}_q{)}^{e_1+e_2}$. Let $Y_i$ be a non-empty subset of $X_i$ and let ${\alpha }_i=\mathrm{\mid }{Y}_i\mathrm{\mid }\mathrm{/}\mathrm{\mid }{X}_i\mathrm{\mid }$. We give a positive lower bound, depending only on ${\alpha }_1,{\alpha }_2,e_1,e_2,q$, for the proportion of pairs $(S_1,S_2)\in Y_1\times Y_2$ which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.