Global eigenvalue distribution of matrices defined by the skew-shift

Abstract

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\left(\begin{array}{c}j\\ 2\end{array}\right)\omega +jy+x\mathrm{mod}1$ 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.