[Paper Review] Toeplitz band matrices with small random perturbations

Johannes Sjoestrand, Martin Vogel|arXiv (Cornell University)|Jan 25, 2019

Random Matrices and Applications15 references4 citations

TL;DR

This paper studies the spectral behavior of large $N\times N$ Toeplitz band matrices under small complex Gaussian random perturbations. It establishes a probabilistic Weyl law and proves that, with high probability, most eigenvalues lie within $\mathcal{O}(N^{-1+\varepsilon})$ of the symbol curve $p(S^1)$, demonstrating eigenvalue concentration near the spectral curve despite non-normality.

ABSTRACT

We study the spectra of $N imes N$ Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on $N$, with probability sub-exponentially (in $N$) close to $1$. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most $\mathcal{O}(N^{-1+\varepsilon})$, for all $\varepsilon >0$, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.

Motivation & Objective

• To understand the spectral distribution of non-self-adjoint Toeplitz band matrices under small random perturbations.
• To establish a probabilistic Weyl law for the number of eigenvalues in domains that may depend on $N$.
• To analyze the asymptotic localization of eigenvalues relative to the symbol curve $p(S^1)$ in the complex plane.
• To show that random perturbations suppress spurious eigenvalues and restore spectral concentration near the essential spectrum.

Proposed method

• Use of the Grushin problem framework to analyze the resolvent and eigenvalue counting of perturbed Toeplitz operators.
• Application of a probabilistic Weyl law via large deviation estimates and Borel-Cantelli arguments to control eigenvalue distribution.
• Employment of the logarithmic potential $U_{\mu}(z) = -\frac{1}{2\pi}\int \log|z-w|\,d\mu(w)$ to relate eigenvalue counting to the Green's function of the limiting measure.
• Analysis of the determinant $\det(P_N^\delta - z)$ via asymptotic expansions and estimates on the off-diagonal block $E_{-+}^\delta(z)$ in the Grushin framework.
• Use of the Hilbert-Schmidt norm and Gaussian random matrix theory to model small perturbations $\delta Q_\omega$ with i.i.d. complex Gaussian entries.
• Establishment of convergence of the empirical spectral measure $\xi_N$ to the pushforward measure $\xi = p_*\left(\frac{1}{2\pi}L_{S^1}\right)$ via convergence of logarithmic potentials in $\mathcal{D}'$.

Experimental results

Research questions

• RQ1 How does the spectrum of a large non-self-adjoint Toeplitz band matrix behave under small random perturbations?
• RQ2 What is the asymptotic distribution of eigenvalues in the complex plane when $N \to \infty$ and the perturbation $\delta \ll 1$?
• RQ3 Can a probabilistic Weyl law be established for the number of eigenvalues in domains depending on $N$?
• RQ4 To what extent do eigenvalues of the perturbed matrix concentrate near the symbol curve $p(S^1)$?
• RQ5 How does the empirical spectral measure of the perturbed matrix converge to the spectral measure of the infinite Toeplitz operator?

Key findings

• With probability sub-exponentially close to 1 in $N$, the number of eigenvalues in any domain $\Omega_N$ is asymptotically given by a probabilistic Weyl law.
• All but a negligible fraction of eigenvalues of $P_N^\delta$ lie within $\mathcal{O}(N^{-1+\varepsilon})$ of the curve $p(S^1)$ for any $\varepsilon > 0$.
• The empirical spectral measure $\xi_N$ converges weakly almost surely to the pushforward measure $\xi = p_*\left(\frac{1}{2\pi}L_{S^1}\right)$ as $N \to \infty$.
• The logarithmic potential $U_{\xi_N}(z)$ converges almost surely to $U_\xi(z)$ for almost every $z \in \mathbb{C}$, implying convergence of the spectral measures.
• The operator norm of $P_N^\delta$ is bounded by $\|p\|_{L^\infty(S^1)} + 1$ for large $N$, with high probability.
• The eigenvalue distribution is robust to perturbations: the random matrix $\delta Q_\omega$ suppresses spurious eigenvalues and forces concentration near $p(S^1)$.