论文标题
随机整数矩阵的奇异性
Singularity of random integer matrices with large entries
论文作者
论文摘要
我们研究随机整数矩阵的奇异性概率。具体而言，从$ \ { - m，\ ldots，m \} $均匀选择的整数条目的随机$ n \ times n $矩阵的可能性是单数。这个问题已经在两个方案中进行了很好的研究：大$ n $和常数$ m $；或大型$ m $和常数$ n $。在本文中，我们扩展了以前的技术来处理$ n，m $都大的政权。我们表明，对于某些绝对常数$ c> 0 $，这种矩阵是单数为$ m^{ - cn} $的概率。我们还为编码理论提供了一些结果。
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.
