论文标题
关于随机矩阵的特征多项式的精确偏差
On the precise deviations of the characteristic polynomial of a random matrix
论文作者
论文摘要
在本文中，使用我们先前关于Mod-Gaussis融合理论的技术开发的技术，我们证明了随机单位矩阵的特征多项式对数的精确中等和大偏差结果。在根据HAAR量度选择单位矩阵的情况下，特征多项式的对数的订单$ a = o（n）$的概率的对数估计了Hughes，Keating和O'Connell。在这项工作中，我们给出了概率本身（没有对数）的等效，我们这样做是为任何参数$β> 0 $的圆形$β$集合的矩阵的更一般情况。与Féray-Méliot-Nikeghbali（2016）和Dal Borgo-Hovhannisyan-Rouault（2019）的先前结果相比，我们大大扩展了可以编写精确估计值的波动范围。
In this paper, using techniques developed in our earlier works on the theory of mod-Gaussian convergence, we prove precise moderate and large deviation results for the logarithm of the characteristic polynomial of a random unitary matrix. In the case where the unitary matrix is chosen according to the Haar measure, the logarithms of the probabilities of fluctuations of order $A=O(N)$ of the logarithm of the characteristic polynomial have been estimated by Hughes, Keating and O'Connell. In this work we give an equivalent of the probabilities themselves (without the logarithms), and we do so for the more general case of a matrix from the circular $β$ ensemble for any parameter $β> 0$. In comparison to previous results from Féray-Méliot-Nikeghbali (2016) and Dal Borgo-Hovhannisyan-Rouault (2019), we considerably extend the range of fluctuations for which precise estimates can be written.
