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 j 2 ω + jy + x mod 1 for irrational ω. 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.

On the finite-size Lyapunov exponent for the Schroedinger operator with skew-shift potential

“…(Recall that the latter is only localized for λ > 1.) The observation that the skew-shift is more quasi-random than the shift has been made in another context by Rudnick-Sarnak-Zaharescu [21] and others [13,19,20] (concerning the spacing distribution) and also recently in [1] (concerning eigenvalues of large Hermitian matrices).…”

“…A one-dimensional quantum particle living on Z with energy E ∈ R is described by the discrete Schrödinger equation (1) ψ n+1 + ψ n−1 + λv n ψ n = Eψ n , where ψ = (ψ n ) n∈Z is a sequence in 2 (Z; C). The real-valued potential sequence v = (v n ) n∈Z represents the environment that the particle is subjected to.…”