Thresholds in the Lattice of Subspaces of $(\mathbb F_q)^n$

TL;DR

This paper establishes a sharp threshold phenomenon for the distribution of subspaces in the lattice of $(F_q)^n$, showing how the density at one dimension influences the densities at higher dimensions.

Contribution

It introduces a threshold theorem linking densities of subspaces across dimensions in the lattice of $(F_q)^n$, with explicit bounds and implications.

Findings

Derived a relation between densities at consecutive dimensions

Proved a sharp threshold theorem for subspace densities

Quantified how low density at one dimension bounds higher dimensions

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 \[ \mu_k(Q) = \frac{1}{1+z} \quad\Longrightarrow\quad \mu_{k+1}(Q) \le \frac{1}{1+qz}. \] This implies a sharp threshold theorem: if $\mu_k(Q) \le 1-\varepsilon$, then $\mu_\ell(Q) \le \varepsilon$ for $\ell = k + O(\log_q(1/\varepsilon))$.