Mathematics – Combinatorics
Scientific paper
2010-12-10
Mathematics
Combinatorics
17 pages, no figures, submitted
Scientific paper
Let $A$ be an $n \times n$ random matrix with iid entries over a finite field of order $q$. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - \alpha $ for some fixed $0 < \alpha < 1$. We show that the singularity probability converges to the uniform limit with error bounded by $O(e^{-c\alpha n})$, where the implied constant and $c > 0$ are absolute. We also show that the determinant of $A$ assumes each non-zero value with probability $q^{-1} \prod_{k=2}^\infty (1 - q^{-k}) + O(e^{-c\alpha n})$, where the constants are absolute.
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