Spectral measure of large random Helson matrices

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 ∈ L2+ ε(P) for some ε > 0 and Var(X) = 1. For the Helson matrices generated by the independent copies of X, scaling the eigenvalues by √n, we prove the almost sure weak convergence of the spectral measure to the standard semi-circular law. Similar results are established for large random matrices with certain general patterned structures.