Singularity of random integer matrices with large entries

Abstract

We study the singularity probability of random integer matrices. Concretely, the probability that a random $n \times n$ matrix, with integer entries chosen uniformly from $\{-m,\ldots,m\}$, is singular. This problem has been well studied in two regimes: large $n$ and constant $m$; or large $m$ and constant $n$. In this paper, we extend previous techniques to handle the regime where both $n,m$ are large. We show that the probability that such a matrix is singular is $m^{-cn}$ for some absolute constant $c>0$. We also provide some connections of our result to coding theory.