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Singularity degree of structured random matrices

Torben Krüger, David Renfrew

Abstract

We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up, which we label the singularity degree.

Singularity degree of structured random matrices

Abstract

We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up, which we label the singularity degree.

Paper Structure

This paper contains 8 sections, 16 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Block singularity degree
  4. Singular stability
  5. Min-max averaging problem
  6. Example
  7. Numerics
  8. Non-negative matrices

Key Result

Proposition 2.1

Depending on the support properties of $S$, the self-consistent density of states DOS split has a point mass at zero, a density blow-up or a bounded density. More precisely

Figures (2)

  • Figure 1: Histrogram of eigenvalues of $H$ and solution to the MDE
  • Figure 2: Log-log plots with the size $n$ of blocks of the random matrices along the horizontal axis and the average of the least singular value over 200 trials on the vertical axis

Theorems & Definitions (40)

  • Proposition 2.1: Singularity at zero
  • Lemma 2.2: Normal form of symmetric non-negative matrix
  • Definition 2.3: 0-1 mask
  • Definition 2.4: Complement index
  • Definition 2.5: Order
  • Definition 2.6: Length
  • Lemma 2.7: Well-definedness of $\ell_\LHD(R)$
  • Theorem 2.8: Classification of singularities
  • Remark 2.9
  • Lemma 3.1: Min-max-averaging of indices
  • ...and 30 more