## Stein’s method and the rank distribution of random matrices over finite fields.(English)Zbl 1388.60024

Summary: 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 $$q\geq2$$, and $$\mathcal{Q}_{q}$$ the distributional limit of $$\mathcal{Q}_{q,n}$$ as $$n\to\infty$$, we apply Stein’s method to prove the total variation bound $\frac{1}{8q^{n+m+1}}\leq\parallel\mathcal{Q}_{q,n}-\mathcal{Q}_{q}\parallel_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}.$ In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.

### MSC:

 60B20 Random matrices (probabilistic aspects) 15B52 Random matrices (algebraic aspects) 60C05 Combinatorial probability 60F05 Central limit and other weak theorems

### Keywords:

Stein’s method; random matrix; finite field; rank
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### References:

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