  We consider large Hermitian matrices whose entries are defined by evaluating
the exponential function along orbits of the skew-shift $\binom{j}{2}
\omega+jy+x \mod 1$ 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 quasi-random nature of the skew-shift dynamics which was
observed in other contexts by Bourgain-Goldstein-Schlag and
Rudnick-Sarnak-Zaharescu.
