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Spectral measure of large random Helson matrices

Yanqi Qiu, Guocheng Zhen

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.

Spectral measure of large random Helson matrices

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 for some and . For the random Helson matrices generated by the independent copies of , scaling the eigenvalues by , 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.
Paper Structure (21 sections, 20 theorems, 146 equations, 1 figure)

This paper contains 21 sections, 20 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Outline of the Proofs of Theorem \ref{['thm-main']} and Theorem \ref{['thm-gen-semi']}
  4. Discussion on some assumptions
  5. Preliminaries
  6. The bounded lipschitz metric
  7. A useful rank inequality
  8. Asymptotic of the number of divisors
  9. Numbers of entries in multiplication table
  10. Moment methods
  11. Some combinatorics on partition words and circuits
  12. The properties of partition words and Catalan words
  13. System of equations associated to a partition word and a map $S$
  14. Properties for circuits
  15. Proofs of Theorems.
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X$ be an arbitrary real random variable such that $\mathrm{Var}(X)=1$ and $\mathbb{E}[|X|^{2+\varepsilon}]<\infty$ for some $\varepsilon >0$. Let $H_{n}$ be the corresponding random $n \times n$ Helson matrix. Then, with probability 1, $\widehat{\mu}(H_{n} / \sqrt{n})$ converges weakly to the W

Figures (1)

  • Figure 1: Comparison between $\widehat{\mu}(H_{1000}/\sqrt{1000})$ and the Wigner semi-circular law $\gamma_{sc}$. (generated by standard Gaussian variables and both scaled up by $2\pi$ times)

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark
  • Remark
  • Theorem 1.2
  • Lemma 2.1: From moments convergence to weak convergence
  • Definition
  • Lemma 3.1
  • Lemma 3.2: The structure of the reduced words
  • proof
  • Corollary 3.3: The structure of non-Catalan partition words
  • ...and 29 more