Global eigenvalue distribution of matrices defined by the skew-shift

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of skew-shift $\binom{j}{2} \omega+jy+x \mod 1$ for irrational $\omega$. prove that eigenvalue distribution these converges to corresponding from random matrix theory on global scale, namely, Wigner semicircle law square and Marchenko-Pastur rectangular matrices. The results evidence quasi-random nature dynamics which was observed in other contexts Bourgain-Goldstein-Schlag Rudnick-Sarnak-Zaharescu.