Comment on "Symmetric Pseudo-Random Matrices"

Abstract

In 2018, Soloveychik, Xiang and Tarokh considered a pseudo-random symmetric circulant matrix constructed from binary Golomb sequences of length $n=2^m-1$, and claimed a proof that its empirical spectral distribution converges almost surely to the semicircle law as $n$ grows to infinity by the method of moments. In this comment note we show that their argument contains several technical flaws and inappropriate applications of technical lemmas. Instead we demonstrate that the eigenvalues of the matrix are simply a normalized twisted Kloosterman sum varying over the multiplicative character, to which we apply Katz's result directly to establish the asymptotically semicircle spectral distribution deterministically.