We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift ($\genfrac{}{}{0ex}{}{j}{2}$) $\omega +jy+xmod1$ for irrational $\omega$. We prove that the eigenvalue distribution of these matrices converges to the corresponding distribution from random matrix theory on the global scale, namely, the Wigner semicircle law for square matrices and the Marchenko–Pastur law for rectangular matrices. The results evidence the quasirandom nature of the skew-shift dynamics which was observed in other contexts by Bourgain, Goldstein and Schlag and Rudnick, Sarnak and Zaharescu.

我们考虑大型 Hermitian 矩阵，其条目是通过评估沿偏斜偏移轨道的指数函数来定义的（$\genfrac{}{}{0ex}{}{j}{2}$) $\omega +j是+X模组1$ 为非理性 $\omega$. 我们证明这些矩阵的特征值分布在全局尺度上收敛到随机矩阵理论的相应分布，即方阵的 Wigner 半圆定律和矩形矩阵的 Marchenko-Pastur 定律。结果证明了 Bourgain、Goldstein 和 Schlag 以及 Rudnick、Sarnak 和 Zaharescu 在其他情况下观察到的斜移动力学的准随机性质。