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

2012/11/30 by Jason Fulman, Larry Goldstein · 1 citation
Mathematics · #math.PR #math.CO

paper · pdf · doi:10.1214/13-aop889

published as Annals of Probability 2015, Vol. 43, No. 3, 1274-1314 · Published at http://dx.doi.org/10.1214/13-AOP889 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

arxiv created 2015/05/14 · arxiv updated 2015/05/15

Abstract

With Q_q,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 \mathbbF_q of size q≥2, and Q_q the distributional limit of Q_q,n as n→∞, we apply Stein's method to prove the total variation bound \frac18q^n+m+1≤‖Q_q,n-Q_q‖_TV≤\frac3q^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.

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• Quadratic spaces and Selmer groups of abelian varieties with multiplication

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