Linking effective Ratner equidistribution to the semicircle law for skew-shift matrices

Abstract

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift \(\frac{j(j-1)}{2}ω+ jy + x \mod 1\) for irrational \(ω\). We establish a rigorous connection between the effective Ratner equidistribution theorem for unipotent orbits in \(\SL(3,\R)/\SL(3,\Z)\) and the global semicircle law for such deterministic matrices. For frequency sequences satisfying a Diophantine condition, we prove that the empirical spectral distribution of these matrices converges to the Wigner semicircle law with optimal polynomial rate \(O(N^{-1})\); for rectangular matrices the corresponding Marchenko--Pastur law is obtained. The proof uses a multi-parameter effective mixing property derived from the effective Ratner equidistribution theorem, combined with a graph-theoretic expansion of the moments. Our results evidence the quasirandom nature of the skew-shift dynamics observed in other contexts by Bourgain, Goldstein and Schlag, and Rudnick, Sarnak and Zaharescu, and provide a dynamical systems proof of the semicircle law with an improved convergence rate.