Georgia Institute of Technology College of Sciences

A central question in ergodic theory is whether sequences obtained by sampling along the orbits of a given dynamical system behave similarly to sequences of i.i.d. random variables. Here we consider this question from a spectral-theoretic perspective. Specifically, we study large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift on the 2-torus with irrational frequency. We prove that their global eigenvalue distribution converges to the Wigner semicircle law, a hallmark of random matrix statistics, which evidences the quasi-random nature of the skew-shift dynamics. This is joint work with Arka Adhikari and Horng-Tzer Yau.