Stein's method and the rank distribution of random matrices over finite fields

Abstract

With ${\mathcal{Q}}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\times(n+m)$ matrices over the finite field $\mathbb{F}_q$ of size $q\ge2$, and ${\mathcal{Q}}_q$ the distributional limit of ${\mathcal{Q}}_{q,n}$ as $n\rightarrow\infty$, we apply Stein's method to prove the total variation bound $\frac{1}{8q^{n+m+1}}\leq\|{\mathcal{Q}}_{q,n}-{\mathcal{Q}}_q\|_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}$. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.