CUE特征多项式极值的数值研究
Extreme values of CUE characteristic polynomials: a numerical study
摘要：
本文给出了随机酉矩阵特征多项式极值统计量的系统数值计算结果。特别是，我们调查了一系列最近的progratures和理论结果的启发，类比与理论相关的高斯随机场。这些现象包括与凝固转变有关的现象。我们的数值结果与前人的理论和结果一致，从而支持了前人的理论和结果.我们还在几个方向上超越了以前的调查：我们提供了第一个定量的证据来支持特征多项式的极值和频谱中的大间隙之间的相关性，我们研究了收敛到先前考虑的极限公式的速度，并且我们将先前对CUE的分析扩展到C $β$ E，其对应于允许本征值排斥的程度变为参数.
英文摘要：
We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the Circular Unitary Ensemble (CUE) of Random Matrix Theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the C$β$E which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.
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