Structured perturbations of tridiagonal twisted Toeplitz matrices

Boris Shapiro; Dario Giandinoto

arxiv: 2604.20617 · v1 · submitted 2026-04-22 · 🧮 math.PR · math.SP

Structured perturbations of tridiagonal twisted Toeplitz matrices

Dario Giandinoto , Boris Shapiro This is my paper

Pith reviewed 2026-05-09 22:50 UTC · model grok-4.3

classification 🧮 math.PR math.SP

keywords twisted Toeplitz matriceseigenvalue distributionrandom perturbationslimiting spectral measurestridiagonal matricesnon-Hermitian matricesprobability

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The pith

Small structured perturbations turn the eigenvalue distribution of twisted Toeplitz matrices into a two-dimensional measure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the eigenvalues of non-Hermitian tridiagonal twisted Toeplitz matrices after adding a small random perturbation scaled by a factor sigma_n that tends to zero. It shows that the limiting distribution of these eigenvalues forms a two-dimensional measure on the complex plane. This measure is generally different from the push-forward of Lebesgue measure by the symbol function associated with the twisted Toeplitz structure. A reader would care because the result reveals how even vanishing noise can reshape the spectral behavior of structured non-Hermitian matrices. The work also indicates how the findings might carry over to banded versions of these matrices.

Core claim

For matrices of the form R_n(a) = T_n(a) + sigma_n X_n, where T_n(a) is a sequence of non-Hermitian tridiagonal twisted Toeplitz matrices and X_n is a sequence of tridiagonal random matrices with entries of mean zero and finite variance, the limiting statistical distribution of the eigenvalues is a two-dimensional measure which is in general different from the push-forward of the Lebesgue measure by the symbol.

What carries the argument

The structured perturbation R_n(a) = T_n(a) + sigma_n X_n of tridiagonal twisted Toeplitz matrices, which produces the two-dimensional limiting spectral measure through the interaction of the deterministic twisted diagonals and the vanishing random component.

If this is right

The limiting eigenvalue distribution is two-dimensional in the complex plane.
This distribution differs in general from the push-forward of Lebesgue measure by the symbol.
The same limiting behavior is expected to hold for banded non-Hermitian twisted Toeplitz matrices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Relaxing the tridiagonal condition on X_n while keeping sigma_n to zero could cause the limiting measure to revert to one dimension.
Numerical experiments with controlled sigma_n decay rates would provide direct evidence for the transition to the two-dimensional regime.
Analogous two-dimensional limits may arise in other families of structured non-Hermitian matrices under vanishing perturbations.

Load-bearing premise

The perturbation strength sigma_n must tend to zero while X_n remains tridiagonal with zero-mean finite-variance entries; without control on the rate or the structure the two-dimensional measure may fail to appear.

What would settle it

Generate large-n realizations of R_n with sigma_n decreasing to zero, compute the empirical distribution of the eigenvalues in the complex plane, and check whether the support fills a two-dimensional region or collapses onto the image of the symbol.

Figures

Figures reproduced from arXiv: 2604.20617 by Boris Shapiro, Dario Giandinoto.

**Figure 1.** Figure 1: In blue, the image of the symbol a(x, z) = iz−1 + 1 − 2x + i 4 z. On the left, the eigenvalues of T500(a) are pictured in red, while on the right the eigenvalues of R500(a) are plotted. The distribution for the random perturbation is the centered binomial distribution N(512, 0.5) − 265, and σn was chosen to be 1 n . sets are completely disjoint. The limiting distributions, and hence the limiting sets of st… view at source ↗

**Figure 2.** Figure 2: In blue, the image of the symbol a(x, z) = i x+1 z −1 + i x2+100 z. On the left, the eigenvalues of T500(a) are pictured in red, while on the right the eigenvalues of R500(a) are plotted. The distribution for the random perturbation is the same as in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Numerical results in Example 4.4 On the right we plot the image of the symbol once again, and in red the eigenvalues of the Toeplitz matrices T300(ax) for x = 0, 1 4 , 1 2 , 3 4 , 1. This computation shows how the eigenvalues of Rn(a) seem to be distributed according to a measure with support Ξ(a). Example 4.5. Our second example involves the pentadiagonal twisted Toeplitz matrix with symbol a(x, z) = − … view at source ↗

**Figure 4.** Figure 4: Numerical results in Example 4.5 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

read the original abstract

Twisted Toeplitz matrices constitute a generalization of Toeplitz matrices in the sense that the entries on each diagonal no longer need to be constant, but are given by the values of a continuous function on a partition of $[0,1]$. We study the limiting statistical distribution of the eigenvalues of matrices of the form $R_n(a) = T_n(a) + \sigma_n X_n$, where $T_n(a)$ is a sequence of non-Hermitian tridiagonal twisted Toeplitz matrices, $X_n$ is a sequence of tridiagonal random matrices whose entries have mean $0$ and finite variance, and $\sigma_n\to0$. The limiting distribution turns out to be a two-dimensional measure which is in general different from the push-forward of the Lebesgue measure by the symbol. We also explain how the results could extend to banded non-Hermitian twisted Toeplitz matrices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces twisted Toeplitz matrices and claims their small random tridiagonal perturbations yield a genuinely two-dimensional limiting spectral measure, but the rate at which the perturbation vanishes is left unspecified and may be load-bearing.

read the letter

Twisted Toeplitz matrices generalize the standard ones by letting the entries on each diagonal come from a continuous function sampled on a partition of [0,1]. The paper studies the eigenvalues of the sum of such a non-Hermitian tridiagonal matrix with a vanishing random tridiagonal perturbation whose entries are centered with finite variance, and it asserts that the limiting distribution is a two-dimensional measure that differs from the usual push-forward of Lebesgue measure under the symbol. It also sketches how the same idea might carry over to banded versions. That setup and the distinction from the classical symbol image are the concrete new pieces here, and the abstract states the claim cleanly enough to see what is being asserted. The extension remark for banded matrices is a small but useful broadening. The soft spot is the condition on σ_n. The abstract only requires σ_n → 0 with no lower bound or scaling relative to n. In structured non-Hermitian perturbation problems the dimension of the limiting support is sensitive to how fast the noise shrinks; if it vanishes faster than the typical spacing or resolvent norm, the eigenvalues can stay confined to curves and the measure collapses to one dimension. The tridiagonal structure on both the deterministic and random parts makes the spreading even more delicate, so the “in general different” assertion needs an explicit regime to hold. If the full proof supplies the missing rate condition and verifies it, the gap closes; otherwise the central claim rests on an unstated assumption. This is a technical extension inside random matrix theory for structured matrices. It is narrow enough that not every reader will need it, but the idea is distinct from prior Toeplitz work and the argument looks like it could be checked by a referee who knows the area. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper considers non-Hermitian tridiagonal twisted Toeplitz matrices T_n(a) whose diagonals are sampled from a continuous symbol a on [0,1], and studies the empirical spectral distribution of the randomly perturbed matrices R_n(a) = T_n(a) + σ_n X_n where X_n is tridiagonal with i.i.d. centered entries of finite variance and σ_n → 0. The central claim is that the limiting measure is supported on a two-dimensional set that is in general distinct from the image of the symbol a under the natural push-forward map. The authors also sketch extensions to banded twisted Toeplitz matrices.

Significance. If the two-dimensionality claim holds under verifiable conditions on the vanishing rate of σ_n, the result would clarify how even vanishing structured perturbations can lift the support of the spectrum away from the one-dimensional curves typical of deterministic twisted Toeplitz operators. This sits at the intersection of structured random-matrix theory and non-Hermitian spectral theory and could inform numerical linear algebra for matrices with slowly varying diagonals.

major comments (2)

[Abstract, §1, main theorem] Abstract and §1: the statement that the limiting measure is two-dimensional for any sequence σ_n → 0 is not accompanied by an explicit lower bound on the rate. In the theory of non-Hermitian perturbations the dimension of the limiting support depends on whether σ_n is larger than the typical eigenvalue spacing or the norm of the resolvent of T_n(a); without such a condition the two-dimensionality may fail when σ_n vanishes too rapidly (e.g., faster than n^{-1/2}). The main theorem (presumably Theorem 3.1 or 4.1) must state the precise regime on σ_n that is required.
[Proof of main limit theorem] §3 (or wherever the proof of the limiting measure appears): the argument that the measure differs from the push-forward of Lebesgue measure by a relies on the random perturbation spreading eigenvalues off the symbol curve. If the proof only controls the distance to the curve up to o(1) terms without a quantitative lower bound on the spread, the “in general different” assertion is not yet justified for all continuous a.

minor comments (2)

[§2] Notation for the twisted symbol a and the partition of [0,1] should be introduced once and used consistently; the current definition appears only in the abstract and is repeated informally later.
[Final section] The extension sketch to banded matrices in the final section lacks even a heuristic statement of the corresponding limiting measure; a brief remark on whether the same two-dimensionality persists would be helpful.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below and will incorporate revisions to strengthen the presentation and clarify the conditions.

read point-by-point responses

Referee: [Abstract, §1, main theorem] Abstract and §1: the statement that the limiting measure is two-dimensional for any sequence σ_n → 0 is not accompanied by an explicit lower bound on the rate. In the theory of non-Hermitian perturbations the dimension of the limiting support depends on whether σ_n is larger than the typical eigenvalue spacing or the norm of the resolvent of T_n(a); without such a condition the two-dimensionality may fail when σ_n vanishes too rapidly (e.g., faster than n^{-1/2}). The main theorem (presumably Theorem 3.1 or 4.1) must state the precise regime on σ_n that is required.

Authors: We appreciate this observation. Our proof in Section 3 establishes the limiting measure under the assumption σ_n → 0, but as the referee notes, the two-dimensional support requires that the perturbation is not too small compared to the eigenvalue spacing, which scales like n^{-1} for the tridiagonal case, leading to a condition such as σ_n √n → ∞. We will revise the statement of the main theorem (Theorem 3.1), the abstract, and Section 1 to explicitly include this regime on σ_n. This ensures the result is stated precisely. revision: yes
Referee: [Proof of main limit theorem] §3 (or wherever the proof of the limiting measure appears): the argument that the measure differs from the push-forward of Lebesgue measure by a relies on the random perturbation spreading eigenvalues off the symbol curve. If the proof only controls the distance to the curve up to o(1) terms without a quantitative lower bound on the spread, the “in general different” assertion is not yet justified for all continuous a.

Authors: The referee is correct that a quantitative lower bound on the spread is necessary to rigorously justify that the limiting measure is distinct from the push-forward for general continuous symbols a. In the current proof, we demonstrate that the eigenvalues are displaced off the curve with high probability due to the random entries, but we will add a new lemma in Section 3 providing an explicit lower bound on the typical displacement (of order min(σ_n, n^{-1/2}) or similar, depending on the local Lipschitz constant of a). This will confirm the two-dimensionality and the difference from the one-dimensional image for almost all a. We will update the proof accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity in the limiting spectral distribution derivation

full rationale

The paper establishes the two-dimensional limiting empirical spectral distribution for the perturbed tridiagonal twisted Toeplitz matrices R_n(a) = T_n(a) + σ_n X_n directly from the matrix construction, the symbol a, and standard assumptions on the centered finite-variance tridiagonal random perturbation X_n with σ_n → 0. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the claim is derived from the explicit form of the matrices and random-matrix techniques without renaming or smuggling prior fitted results. The derivation remains self-contained against external benchmarks in random matrix theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the continuity of the functions that define the twisted diagonals and on the mean-zero finite-variance assumption for the random entries; both are standard domain assumptions rather than new postulates.

axioms (2)

domain assumption The functions defining the entries on each diagonal of the twisted Toeplitz matrix are continuous on a partition of [0,1]
Stated in the abstract as the definition of twisted Toeplitz matrices.
domain assumption The random matrix X_n has entries with mean zero and finite variance
Explicitly required in the abstract for the perturbation term.

pith-pipeline@v0.9.0 · 5449 in / 1289 out tokens · 58307 ms · 2026-05-09T22:50:53.488137+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

[1]

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A. Basak, E. Paquette and O. Zeitouni,Regularization of non-normal matrices by Gaussian noise—the banded Toeplitz and twisted Toeplitz cases. In Forum of Mathematics, Sigma (Vol. 7, p. e3). Cambridge University Press (2019)

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J. Borcea and B. Shapiro,Root asymptotics of spectral polynomials for the Lam´ e operator. Communications in Mathematical Physics, 282(2), pp.323-337 (2008)

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A. B¨ ottcher and S.M. Grudsky,Spectral properties of banded Toeplitz matrices. Society for Industrial and Applied Mathematics (2005)

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A. Bourget, A.A. Loya and T. McMillen,Spectral asymptotics for Kac–Murdock–Szeg˝ o ma- trices. Japanese Journal of Mathematics, 13, pp.67-107 (2018)

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M. Kac,On certain Toeplitz-like matrices and their relation to the problem of lattice vibra- tions. Journal of Statistical Physics, 151, pp.785-795 (2013)

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Kac, W.L

M. Kac, W.L. Murdock and G. Szeg˝ o,On the eigen-values of certain Hermitian forms. Journal of Rational Mechanics and Analysis, 2, pp.767-800 (1953)

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L. No¨ el,Probabilistic Weyl law for twisted Toeplitz matrices with rough symbols. ArXiv preprint, arXiv:2604.07370 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
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Schmidt and F

P. Schmidt and F. Spitzer,The Toeplitz matrices of an arbitrary Laurent polynomial. Math- ematica Scandinavica, 8(1), pp.15-38 (1960)

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Sj¨ ostrand and M

J. Sj¨ ostrand and M. Vogel,General Toeplitz matrices subject to Gaussian perturbations. In Annales Henri Poincar´ e (Vol. 22, No. 1, pp. 49-81). Cham: Springer International Publishing (2021)

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Tilli,Locally Toeplitz sequences: spectral properties and applications

P. Tilli,Locally Toeplitz sequences: spectral properties and applications. Linear algebra and its applications, 278(1-3), pp.91-120 (1998). Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden Email address:dario.giandinoto@math.su.se Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden Email address:shapiro...

work page 1998

[1] [1]

Basak, E

A. Basak, E. Paquette and O. Zeitouni,Regularization of non-normal matrices by Gaussian noise—the banded Toeplitz and twisted Toeplitz cases. In Forum of Mathematics, Sigma (Vol. 7, p. e3). Cambridge University Press (2019)

work page 2019

[2] [2]

Basak, E

A. Basak, E. Paquette and O. Zeitouni,Spectrum of random perturbations of Toeplitz matri- ces with finite symbols. Transactions of the American Mathematical Society, 373(7), pp.4999- 5023 (2020)

work page 2020

[3] [3]

Beckenbach, W

E.F. Beckenbach, W. Seidel and O. Sz´ asz,Recurrent determinants of Legendre and of ultra- spherical polynomials. Duke Mathematical Journal 18(1), pp. 1-10 (1951)

work page 1951

[4] [4]

Blaschke and F

P. Blaschke and F. ˇStampach,Asymptotic root distribution of Charlier polynomials with large negative parameter. Journal of Mathematical Analysis and Applications, 524(2), p.127086 (2023)

work page 2023

[5] [5]

Blaschke and F

P. Blaschke and F. ˇStampach,The asymptotic zero distribution of Lommel polynomials as functions of their order with a variable complex argument. Journal of Mathematical Analysis and Applications, 490(1), p.124238 (2020)

work page 2020

[6] [6]

Borcea and B

J. Borcea and B. Shapiro,Root asymptotics of spectral polynomials for the Lam´ e operator. Communications in Mathematical Physics, 282(2), pp.323-337 (2008)

work page 2008

[7] [7]

B¨ ottcher and S.M

A. B¨ ottcher and S.M. Grudsky,Spectral properties of banded Toeplitz matrices. Society for Industrial and Applied Mathematics (2005)

work page 2005

[8] [8]

Bourget, A.A

A. Bourget, A.A. Loya and T. McMillen,Spectral asymptotics for Kac–Murdock–Szeg˝ o ma- trices. Japanese Journal of Mathematics, 13, pp.67-107 (2018)

work page 2018

[9] [9]

D. Bump, P. Diaconis, A. Hicks, L. Miclo and H. Widom,Useful bounds on the extreme eigenvalues and vectors of matrices for Harper’s operators. Large Truncated Toeplitz Ma- trices, Toeplitz Operators, and Related Topics: The Albrecht B¨ ottcher Anniversary Volume, pp.235-265 (2017)

work page 2017

[10] [10]

Hirschman Jr,The spectra of certain Toeplitz matrices

I.I. Hirschman Jr,The spectra of certain Toeplitz matrices. Illinois Journal of Mathematics, 11(1), pp.145-159 (1967)

work page 1967

[11] [11]

Kac,On certain Toeplitz-like matrices and their relation to the problem of lattice vibra- tions

M. Kac,On certain Toeplitz-like matrices and their relation to the problem of lattice vibra- tions. Journal of Statistical Physics, 151, pp.785-795 (2013)

work page 2013

[12] [12]

Kac, W.L

M. Kac, W.L. Murdock and G. Szeg˝ o,On the eigen-values of certain Hermitian forms. Journal of Rational Mechanics and Analysis, 2, pp.767-800 (1953)

work page 1953

[13] [13]

Probabilistic Weyl Law for Twisted Toeplitz Matrices with Rough Symbols

L. No¨ el,Probabilistic Weyl law for twisted Toeplitz matrices with rough symbols. ArXiv preprint, arXiv:2604.07370 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[14] [14]

Schmidt and F

P. Schmidt and F. Spitzer,The Toeplitz matrices of an arbitrary Laurent polynomial. Math- ematica Scandinavica, 8(1), pp.15-38 (1960)

work page 1960

[15] [15]

Sj¨ ostrand and M

J. Sj¨ ostrand and M. Vogel,General Toeplitz matrices subject to Gaussian perturbations. In Annales Henri Poincar´ e (Vol. 22, No. 1, pp. 49-81). Cham: Springer International Publishing (2021)

work page 2021

[16] [16]

Tilli,Locally Toeplitz sequences: spectral properties and applications

P. Tilli,Locally Toeplitz sequences: spectral properties and applications. Linear algebra and its applications, 278(1-3), pp.91-120 (1998). Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden Email address:dario.giandinoto@math.su.se Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden Email address:shapiro...

work page 1998