Thresholds in the Lattice of Subspaces of $({\mathbb{F}}_q{)}^n$

Abstract

Let $Q$ be an ideal (downward-closed set) in the lattice of linear subspaces of $({\mathbb{F}}_q{)}^n$, ordered by inclusion. For $0\le k\le n$, let ${\mu }_k(Q)$ denote the fraction of $k$-dimensional subspaces that belong to $Q$. We show that these densities satisfy \[ μ_k(Q) = \frac{1}{1+z} \quad\Longrightarrow\quad μ_{k+1}(Q) \le \frac{1}{1+qz}. \] This implies a sharp threshold theorem: if ${\mu }_k(Q)\le 1-\epsilon$, then ${\mu }_{\ell }(Q)\le \epsilon$ for $\ell =k+O({\log }_q(1/\epsilon ))$.