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On the operator norm of a Hermitian random matrix with correlated entries

Jana Reker

Abstract

We consider a correlated $N\times N$ Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one.

On the operator norm of a Hermitian random matrix with correlated entries

Abstract

We consider a correlated Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one.

Paper Structure

This paper contains 10 sections, 12 theorems, 80 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation and Assumptions on the Model
  3. Statement of the Main Result
  4. Setup for the Moment Method
  5. Visualization Using Graphs
  6. Proof of \ref{['est-by-const']} for Terms Involving Only 2-Cumulants
  7. Proof of Lemma \ref{['2-cumu-lemma']}
  8. Proof of \ref{['est-by-const']} in the General Case
  9. Including 3-cumulants
  10. Including Cumulants of Order Four and Higher

Key Result

Theorem 2

Under the assumptions A1-A3, we have that for all ${{\varepsilon}>0}$, ${D>0}$ there exists a suitable constant $C({\varepsilon},D)$ such that, for all $N\in\mathds{N}$,

Figures (6)

  • Figure 1: Visualization of $\kappa^{(2)}(a_1a_2,a_5a_6)\kappa^{(2)}(a_2a_3,a_6a_7)\kappa^{(2)}(a_3a_4,a_7a_8)\kappa^{(2)}(a_4a_5,a_8a_1)$
  • Figure 2: A crossing (left) and a non-crossing pairing (right) for $k=10$.
  • Figure 3: The steps of the reduction algorithm with arrows indicating the steps.
  • Figure 4: Visualization of the rewriting and summing in for a crossing partition.
  • Figure 5: The only subgraphs not covered by previous estimates.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Definition 1: Cumulants
  • Theorem 2
  • Remark
  • Definition 3
  • Definition 4
  • Lemma 5
  • Lemma 6
  • Example 7
  • Definition 8
  • Lemma 9
  • ...and 21 more