有限域上随机矩阵的一些新结果
Some new results in random matrices over finite fields
摘要：
本文给出了与模素数p的均匀分布有较大差异的可能不同步长的随机游动的各种特征，并利用这些结果研究了F_p上随机矩阵秩的分布和随机超平面的法向量的等分布性态。我们还研究了一个随机方阵是特征值自由的概率，或当它的特征多项式是由一个给定的不可约多项式整除在极限n到无穷大的F_p。我们表明，这些统计量是普遍的，扩展Stong和Neumann-Praeger的结果超越了统一的模型。
英文摘要：
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 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 infinity in F_p. We show that these statistics are universal, extending results of Stong and Neumann-Praeger beyond the uniform model.
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