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arXiv · 1903.11514

Global eigenvalue distribution of matrices defined by the skew-shift

Arka Adhikari·Marius Lemm·Horng-Tzer Yau
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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.

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Arka Adhikari, Marius Lemm, Horng-Tzer Yau. 2019-03-27. Global eigenvalue distribution of matrices defined by the skew-shift. https://doi.org/10.2140/apde.2021.14.1153

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