[Paper Review] Asymptotic eigenvalue distribution of large Toeplitz matrices

Seung‐Yeop Lee, Hui Dai+1 more|ArXiv.org|Aug 23, 2007

Random Matrices and Applications12 references3 citations

TL;DR

This paper derives the asymptotic eigenvalue distribution of large Toeplitz matrices with a single jump singularity in the symbol, using the Fisher-Hartwig theorem to analyze the determinant det(ζ − Tₙ(a)). It shows that eigenvalues converge to the symbol's image with deviations scaling as 1/n and log n/n, with non-universal corrections near spectral edges due to logarithmic divergences.

ABSTRACT

We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how the eigenvalues converge to the image. For a given Toeplitz matrix $T_n(a)$ of size $n$, we take the standard approach of looking at $\det(ζ-T_n(a))$, of which the asymptotic information is given by the Fisher-Hartwig theorem. For a symbol with single jump, we obtain the distribution of eigenvalues as an expansion involving $1/n$ and $\log n/n$. To demonstrate the validity of our result we compare our result against the numerics using a pure Fisher-Hartwig symbol.

Motivation & Objective

• To understand the asymptotic distribution of eigenvalues of large Toeplitz matrices generated by singular symbols.
• To resolve how eigenvalues approach the image of the symbol as matrix size n → ∞, particularly for symbols with a single jump discontinuity.
• To provide a quantitative expansion of eigenvalue deviations from the symbol's image, including subleading corrections.
• To validate the analytical results against numerical computations using a pure Fisher-Hartwig symbol.

Proposed method

• Applies the Fisher-Hartwig theorem to the characteristic polynomial det(ζ − Tₙ(a)) to extract asymptotic spectral information.
• Expands the logarithmic derivative of the determinant in powers of 1/n and log n/n to capture eigenvalue deviations.
• Uses Hilbert transform techniques on logarithmic symbols to compute spectral measures and eigenvalue density corrections.
• Derives an explicit formula for the asymptotic spectral measure of symbols with a single jump singularity.
• Compares analytical predictions with numerically computed eigenvalues for a pure Fisher-Hartwig symbol.
• Analyzes edge effects by examining the behavior of δa(e^{iθ}) near θ = ±π, revealing logarithmic divergences in the imaginary part.

Experimental results

Research questions

• RQ1 How do eigenvalues of large Toeplitz matrices with a single jump singularity converge to the image of the symbol?
• RQ2 What is the functional form of the leading-order corrections to the eigenvalue distribution beyond the leading asymptotic behavior?
• RQ3 How do eigenvalue deviations scale with matrix size n, and what role do logarithmic terms play?
• RQ4 Why do eigenvalues near the spectral edges exhibit qualitatively different distribution patterns compared to bulk eigenvalues?
• RQ5 To what extent does the uniform distribution of eigenvalues in θ-space hold, and what constraints does it impose on the perturbative expansion?

Key findings

• The asymptotic eigenvalue distribution is given by an expansion in 1/n and log n/n, with the leading correction terms derived explicitly for a single-jump symbol.
• The deviation of eigenvalues from the symbol's image is governed by a line of discontinuity in det(ζ − Tₙ(a)), which captures the spectral density corrections.
• Near θ = π, the imaginary part of the deviation δa(e^{iθ}) exhibits a logarithmic divergence of the form ∼ (log n/n) log(π − θ), leading to a hook-shaped structure in the eigenvalue distribution.
• The competition between log n/n and 1/n terms determines the dominance of different correction terms, with log n/n dominating when |log(π − θ)| ≪ log n.
• The eigenvalue nearest to the spectral edge scales as ∼ (log log n)(log n)/n, indicating a non-universal, enhanced deviation near the endpoints.
• Numerical results for a pure Fisher-Hartwig symbol show excellent agreement with the analytical predictions, validating the perturbative framework.