Spectral measure of large random Helson matrices

Abstract

We study the limiting spectral measure of large random Helson matrices and large random matrices of certain patterned structures. Given a real random variable $X \in L^{2+ \varepsilon}(\mathbb{P}) $ for some $\varepsilon > 0$ and $\mathrm{Var}(X) = 1$. For the random $n \times n$ Helson matrices generated by the independent copies of $X$, scaling the eigenvalues by $\sqrt{n}$, we prove the almost sure weak convergence of the spectral measure to the standard Wigner semi-circular law. Similar results are established for large random matrices with certain general patterned structures.