Vol. 14, No. 4, 2021

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Global eigenvalue distribution of matrices defined by the skew-shift

Arka Adhikari, Marius Lemm and Horng-Tzer Yau

Vol. 14 (2021), No. 4, 1153–1198

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 quasirandom nature of the skew-shift dynamics which was observed in other contexts by Bourgain, Goldstein and Schlag and Rudnick, Sarnak and Zaharescu.

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random matrices, semicircle law, skew-shift dynamics
Mathematical Subject Classification 2010
Primary: 60B20
Secondary: 37A50, 60F05
Received: 12 April 2019
Revised: 30 August 2019
Accepted: 2 December 2019
Published: 6 July 2021
Arka Adhikari
Department of Mathematics
Harvard University
Cambridge, MA
United States
Marius Lemm
Department of Mathematics
Harvard University
Cambridge, MA
United States
Horng-Tzer Yau
Department of Mathematics
Harvard University
Cambridge, MA
United States