Spectral measure of large random Helson matrices

TL;DR

This paper investigates the spectral distribution of large random Helson matrices and patterned matrices, demonstrating convergence to the Wigner semi-circular law under certain conditions.

Contribution

It proves the almost sure weak convergence of spectral measures of large random Helson matrices to the semi-circular law, extending results to other patterned matrices.

Findings

Spectral measure converges to Wigner semi-circular law

Results hold for matrices generated by independent copies of a random variable

Applicable to large matrices with certain structured patterns

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.