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+ \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.

💡 Research Summary

This paper investigates the limiting spectral distribution of large random Helson matrices and, more generally, of random matrices whose entries follow a prescribed deterministic pattern. A Helson matrix is defined by the multiplicative rule (H_n(i,j)=X_{ij}=X_{i\cdot j}), where ({X_k}{k\ge1}) are i.i.d. copies of a real random variable (X) with mean zero, variance one and a finite ((2+\varepsilon))-moment for some (\varepsilon>0). The main result (Theorem 1.1) shows that, after scaling the matrix by (\sqrt{n}), the empirical spectral measure (\mu(H_n/\sqrt{n})) converges almost surely, in the weak sense, to the standard Wigner semi‑circular law (\gamma{\mathrm{sc}}) supported on (