[Paper Review] Interior eigenvalue density of large bi-diagonal matrices subject to random perturbations

Johannes Sjoestrand, Martin Vogel|arXiv (Cornell University)|Apr 19, 2016

Random Matrices and Applications12 references3 citations

TL;DR

This paper analyzes the average eigenvalue density of large bi-diagonal Toeplitz matrices perturbed by small Gaussian noise. Using complex analysis and asymptotic methods, it derives a precise formula for the eigenvalue intensity in the interior of the convex hull of the symbol's range, showing it converges to a smooth density governed by the logarithmic derivative of a generating function, with explicit error bounds under scaling conditions on the perturbation strength.

ABSTRACT

We study the spectrum of large a bi-diagonal Toeplitz matrix subject to a Gaussian random perturbation with a small coupling constant. We obtain a precise asymptotic description of the average density of eigenvalues in the interior of the convex hull of the range symbol.

Motivation & Objective

• To understand the spectral instability of non-self-adjoint operators, particularly bi-diagonal Toeplitz matrices, under small random perturbations.
• To resolve the lack of detailed description of eigenvalues inside the convex hull of the symbol's range, which previous methods only bounded logarithmically.
• To derive a sharp asymptotic formula for the average eigenvalue density in the interior of confocal ellipses associated with the perturbed matrix.
• To establish a probabilistic Weyl-type law for eigenvalues in the interior region, complementing prior results on the boundary.

Proposed method

• The authors model the perturbed matrix as $ P_\delta = P + \delta Q_\omega $, where $ Q_\omega $ is a complex Gaussian random matrix with i.i.d. entries.
• They study the first intensity measure $ \nu $ of the random eigenvalue counting measure $ \Xi $, defined via $ \mathbb{E}[\Xi(\varphi)] = \int \varphi(z) \nu(dz) $.
• Using complex analysis and subharmonic estimates, they derive an asymptotic expression for the eigenvalue intensity $ \xi(z) $ in terms of the logarithmic derivative of the generating function $ K_\infty(z) $, specifically $ \xi(z) = \frac{2}{\pi} \partial_z \partial_{\bar{z}} \ln K_\infty(z) + \text{error} $.
• The error terms are controlled via estimates on $ |\zeta_-|^{N-1} $, $ F_N $, and the coupling constant $ \delta $, under the assumption $ \delta F_N \gg N^{-2} $ and $ \delta N^3 + \frac{N|\zeta_-|^{N-1}}{\delta F_N^2} \ll 1 $.
• The method builds on techniques from random matrix theory and semiclassical analysis, particularly adapting approaches from [22] for differential operators.
• Key components include the use of Fourier analysis on $ \ell^2(\mathbb{Z}) $, the symbol $ p(\xi) = a e^{i\xi} + b e^{-i\xi} $, and the identification of confocal ellipses as level sets of the numerical range.

Experimental results

Research questions

• RQ1 How does the average eigenvalue density behave in the interior of the convex hull of the symbol's range for large bi-diagonal Toeplitz matrices under small random perturbations?
• RQ2 Can a precise asymptotic formula be derived for the eigenvalue intensity in the interior region, beyond the logarithmic bounds of prior work?
• RQ3 What is the role of confocal ellipses in organizing the eigenvalue distribution under such perturbations?
• RQ4 How do the scaling parameters $ \delta $ and $ N $ affect the convergence of the eigenvalue density to the limiting form?
• RQ5 To what extent does the eigenvalue intensity in the interior depend on the geometry of the symbol's range and the spectral parameters $ a, b $?

Key findings

• The average eigenvalue density $ \nu $ has a continuous Lebesgue density in the interior of the convex hull of the symbol's range, given asymptotically by $ \frac{2}{\pi} \partial_z \partial_{\bar{z}} \ln K_\infty(z) $.
• The leading-order term of the eigenvalue intensity is $ \frac{2}{\pi} \partial_z \partial_{\bar{z}} \ln K_\infty(z) $, which is proportional to the curvature of the generating function $ K_\infty(z) $.
• The error in the asymptotic approximation is bounded by $ \mathcal{O}(F_N^2)\left( \frac{N|\zeta_-|^{N-1}}{\delta F_N^2} + \delta N^3 \right) $, which vanishes under the scaling condition $ \left( \frac{N|\zeta_-|^{N-1}}{\delta}(1-|\zeta_-|)^2 + \delta N^3 \right) \ll 1 $.
• For $ z $ in the interior region $ \Sigma_{r_0 - 1/N} \setminus \Sigma_{r_1} $, the intensity converges to the smooth density $ \frac{2}{\pi} \partial_z \partial_{\bar{z}} \ln K_\infty(z) $ with high probability as $ N \to \infty $.
• The result confirms a refined Weyl law for eigenvalues in the interior, extending previous results that only described the boundary distribution.
• The analysis shows that the eigenvalue density is governed by the geometry of the confocal ellipse family $ p_{ra, r^{-1}b}(\mathbb{R}) $, with $ r \in [(|b|/|a|)^{1/2}, \infty) $.