Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Outliers of perturbations of banded Toeplitz matrices

Charles Bordenave, François Chapon, Mireille Capitaine

Abstract

Toeplitz matrices form a rich class of possibly non-normal matrices whose asymptotic spectral analysis in high dimension is well-understood. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this work, we analyze the spectrum of a banded Toeplitz matrix perturbed by a random matrix with iid entries of variance $σ_n^2 / n$ in the asymptotic of high dimension and $σ_n$ converging to $σ\geq 0$. Our results complement and provide new proofs on recent progresses in the case $σ= 0$. For any $σ\geq 0$, we show that the point process of outlier eigenvalues is governed by a low-dimensional random analytic matrix field, typically Gaussian, alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. On our way, we prove a new functional central limit theorem for trace of polynomials in deterministic and random matrices and present new variations around Szego's strong limit theorem.

Outliers of perturbations of banded Toeplitz matrices

Abstract

Toeplitz matrices form a rich class of possibly non-normal matrices whose asymptotic spectral analysis in high dimension is well-understood. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this work, we analyze the spectrum of a banded Toeplitz matrix perturbed by a random matrix with iid entries of variance in the asymptotic of high dimension and converging to . Our results complement and provide new proofs on recent progresses in the case . For any , we show that the point process of outlier eigenvalues is governed by a low-dimensional random analytic matrix field, typically Gaussian, alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. On our way, we prove a new functional central limit theorem for trace of polynomials in deterministic and random matrices and present new variations around Szego's strong limit theorem.

Paper Structure

This paper contains 48 sections, 41 theorems, 424 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Perturbation of banded Toeplitz matrices
  3. Assumptions on the noise matrix
  4. Convergence of the empirical spectral distribution
  5. Stable region
  6. Unstable region
  7. Banded Toeplitz matrices as perturbed circulant matrices.
  8. The master matrix field.
  9. Convergence of outlier eigenvalues.
  10. Overview of the proofs
  11. Step 1: Sylvester's determinant identity.
  12. Step 2: Taylor expansion for the determinant.
  13. Step 3: Case $\delta = 0$ and Szegő's strong limit theorem.
  14. Step 4: case $\delta \ne 0$, identification of the leading term of the Taylor expansion \ref{['dev-jacobi2']}.
  15. Step 5: convergence of the matrix field.
  16. ...and 33 more sections

Key Result

Lemma 1.1

For $\sigma \geq 0$, the support of $\beta_\sigma$ is given by where by convention $1/0 = \infty$, so that $\mathop{\mathrm{supp}}\nolimits(\beta_0) = \mathop{\mathrm{\bf a}}\nolimits (\mathbb S^1)$.

Figures (3)

  • Figure 1.1: Output for eigenvalues in Python using numpy.linalg.eig for the nilpotent matrix $\mathop{\mathrm{\bf a}}\nolimits(\lambda) = \lambda$. (a)-(b): $S$ is a (random) permutation matrix which hides the upper-triangular structure, and (c)-(d) $F$ is the Fourier basis, see \ref{['eq:defFn']}, which introduces rounding errors in matrix entries.
  • Figure 1.2: Example of the closed curve $\mathop{\mathrm{\bf a}}\nolimits(\mathbb S^1)$ and the support of the Brown measure $\beta_\sigma$, for the symbol $\mathop{\mathrm{\bf a}}\nolimits(t)=t^2+2t+it^{-2}-0.5it^{-3}$ and $\sigma=0.8$.
  • Figure 1.3: Eigenvalues (blue dots) of $T_n(\mathop{\mathrm{\bf a}}\nolimits)+\sigma \frac{X_n}{\sqrt n}$ for $n=2000$, $\sigma=0.6$ and various symbols: (a) $\mathop{\mathrm{\bf a}}\nolimits(t)=t^2$, (b) $\mathop{\mathrm{\bf a}}\nolimits(t)=t+2t^2+t^{-1}$ and (c) $\mathop{\mathrm{\bf a}}\nolimits(t)=-\frac{3}{4} t + \frac{5}{4} i t^2 + \frac{3}{4} t^3 + \frac{3}{4}t^{-1} - i t^{-2} -\frac{3}{4} t^{-3}$. Note that there are no outliers in the bounded region of zero winding number in subfigure (c).

Theorems & Definitions (78)

  • Lemma 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Remark 2.3
  • Proposition 4.1
  • ...and 68 more