Resumen de 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 18qn+m+1≤∥Qq,n−Qq∥TV≤3qn+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.
