# Global eigenvalue distribution of matrices defined by the skew-shift

## TL;DR

This paper proves that large Hermitian matrices constructed from skew-shift orbits exhibit eigenvalue distributions converging to classical random matrix laws, highlighting the quasi-random behavior of skew-shift dynamics.

## Contribution

It establishes the global eigenvalue distribution convergence for matrices defined by skew-shift orbits, connecting dynamical systems with random matrix theory.

## Key findings

- Eigenvalue distribution converges to Wigner semicircle law for square matrices.
- Eigenvalue distribution converges to Marchenko-Pastur law for rectangular matrices.
- Supports the quasi-random nature of skew-shift dynamics.

## Abstract

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\binom{j}{2} \omega+jy+x \mod 1$ 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.