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

With Qq,n the distribution of n minus the rank of a matrix chosen uniformly from the collection of all n×(n+m) matrices over the finite field Fq of size q≥2, and Qq the distributional limit of Qq,n as n→∞, we apply Stein’s method to prove the total variation bound \[{1}{8qⁿ⁺ᵐ⁺¹}≤\|Qq,n-Qq\|TV≤{3}{qⁿ⁺ᵐ⁺¹}.\] In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.