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 \binom{j}{2}\omega+j*y+x mod 1 for irrational frequency \omega. We prove that the global eigenvalue distribution of these matrices converges to the corresponding distributions from random matrix theory, namely, the Wigner semicircle law for square matrices and the Marchenko-Pastur law for rectangular matrices. The result evidences the quasi-random nature of the skew-shift dynamics. This is joint work with Arka Adhikari and Horng-Tzer Yau.