Spectral measure of large random Helson matrices

2024/05/29 by Qiu, Yanqi, Zhen, Guocheng
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2405.18796

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 ∈ L^{2+ ε}(ℙ) for some ε > 0 and Var(X) = 1. For the random n × n 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 Wigner semi-circular law. Similar results are established for large random matrices with certain general patterned structures.

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