In this note, we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime $p$, and use these results to study the distribution of the rank of random matrices over ${\mathbb{F}}_p$ and the equi-distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix is eigenvalue-free, or when its characteristic polynomial is divisible by a given irreducible polynomial in the limit $n\to \infty$ in ${\mathbb{F}}_p$. We show that these statistics are universal, extending results of Stong and Neumann-Praeger beyond the uniform model.

中文翻译：

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有限域上随机矩阵的一些新结果

在本笔记中，我们给出了随机游走的各种特征，这些随机游走可能具有不同的步骤，这些步骤与均匀分布的模数 a 有较大差异 $磷$，并使用这些结果来研究随机矩阵的秩在 ${\mathbb{F}}_{磷}$以及随机超平面法向量的等分布行为。我们还研究了随机方阵没有特征值的概率，或者当它的特征多项式在极限范围内被给定的不可约多项式整除时$n\to \infty$ 在 ${\mathbb{F}}_{磷}$. 我们表明这些统计数据是通用的，将 Stong 和 Neumann-Praeger 的结果扩展到统一模型之外。