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 $X\in L^{2+\epsilon }(\mathbb{P})$ for some $\epsilon >0$ and $\mathrm{V}\mathrm{a}\mathrm{r}(X)=1$. 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.