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arxiv: 2604.07370 · v1 · submitted 2026-04-07 · 🧮 math.PR · math-ph· math.MP· math.SP

Probabilistic Weyl Law for Twisted Toeplitz Matrices with Rough Symbols

Lucas No\"el (IRMA) This is my paper

Pith reviewed 2026-05-10 19:15 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MPmath.SP
keywords Weyl lawToeplitz matricesempirical spectral measurerandom perturbationsHölder continuous symbolsweak convergence in probabilityjump discontinuities
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The pith

Small random perturbations make the spectral measure of twisted Toeplitz matrices converge to the symbol's pushforward even with jumps.

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

The paper establishes that for twisted Toeplitz matrices with symbols that are smooth in frequency but only piecewise Hölder continuous in position with jump discontinuities, adding small random perturbations causes the empirical spectral measure to converge weakly in probability to the measure induced by the symbol. This convergence is a probabilistic Weyl law that holds despite the roughness of the symbol. A sympathetic reader would care because it allows spectral asymptotics to apply to operators with abrupt changes, which are common in physical models. The proof relies on controlling the perturbations to mitigate the effects of the discontinuities.

Core claim

The empirical spectral measure of twisted Toeplitz matrices subject to small random perturbations converges weakly in probability to the push-forward of the Lebesgue measure by the symbol. The symbol is assumed to be smooth in frequency and piecewise Hölder continuous with respect to the position variable with discontinuities of jump type.

What carries the argument

Small random perturbations that smooth the effect of jump discontinuities in the position variable of the symbol while preserving the overall distribution.

If this is right

  • The eigenvalues distribute according to the values taken by the symbol.
  • This holds in probability, so with high probability the distribution matches for large matrices.
  • The result applies to symbols with jump discontinuities that would otherwise prevent standard Weyl laws.
  • Convergence is weak, meaning integrals against continuous test functions converge.

Where Pith is reading between the lines

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

  • This suggests that adding noise can enable Weyl-type laws for a broader class of non-smooth symbols.
  • It may be possible to relax the Hölder condition further or apply to other matrix ensembles.
  • Such results could inform the design of numerical methods for eigenvalue problems with discontinuous coefficients.

Load-bearing premise

The perturbations must be small and the symbol must have the given smoothness in frequency and piecewise Hölder continuity with jumps in position.

What would settle it

A specific example of a symbol with a jump discontinuity where the empirical spectral measure of the perturbed matrix fails to converge to the pushforward of Lebesgue measure by that symbol.

Figures

Figures reproduced from arXiv: 2604.07370 by Lucas No\"el (IRMA).

Figure 1
Figure 1. Figure 1: Graph of f given by (1.14). The symbol p defined in (1.13) belongs to C 0,1/2 pw S1. As mentioned in (1.9), p ver￾ifies Assumption 1. Since f is piecewise H¨older continuous on [0, 1], Proposition B.3 ensures that Assumption 2 holds for p, with κ2 = 1/4. Theorem 1.2 then applies [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The left hand side presents the spectrum of MN (p) without any randomness, for p given by (1.13), and on the right hand side, the spectrum of MN (p) + δQN where QN is a random Gaussian matrix, N = 4000 and δ = N −1.6 . Example 1.5. We now turn to illustrate Corollary 1.3. We consider the pertur￾bative Jordan bloc, say JeN given by JeN :=   0 1 0 0 · · · 0 0 0 1 0 . . . . . . 0 0 0 1 . . . . . . … view at source ↗
Figure 3
Figure 3. Figure 3: is a numerical simulation of the spectrum of JeN + δQN . We can see that the spectrum fills up the numerical range of p, that is S 1 , even with the deterministic perturbation EN,1. We decided to take 2EN,1 instead of EN,1 to show that the eigenvalue 2 seems still present in the spectrum of JeN + δQN , even if most of its eigenvalues are on S 1 as predicted [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

In this article, we study the convergence of the empirical spectral measure of twisted Toeplitz matrices subject to small random perturbations. We show that the empirical spectral measure converges weakly in probability to the push-forward of the Lebesgue measure by the symbol. The symbol of the twisted Toeplitz matrices is assumed to be smooth in frequency, and only piecewise H{\"o}lder continuous with respect to the position variable with discontinuities of jump type.

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.

Referee Report

1 major / 3 minor

Summary. The paper proves that the empirical spectral measure of twisted Toeplitz matrices with symbols smooth in the frequency variable and piecewise Hölder continuous in the position variable (with jump discontinuities), when subjected to sufficiently small random perturbations, converges weakly in probability to the push-forward of Lebesgue measure by the symbol function. This is presented as a probabilistic extension of the Szegő theorem to rough symbols.

Significance. If the proof is complete, the result is significant because it provides a mechanism to obtain Weyl-type laws for non-smooth symbols by adding controlled randomness, which is a novel approach in the intersection of Toeplitz operator theory and random matrix theory. The conditions on the symbol and perturbations appear tailored to control the effect of jumps, potentially opening avenues for further probabilistic spectral results.

major comments (1)
  1. [Main Theorem / §2] The main convergence theorem (presumably Theorem 1.1 or equivalent in §2) states convergence 'in probability' but the proof sketch in the abstract and introduction does not explicitly bound the probability of deviation in terms of the perturbation size; a quantitative estimate (e.g., via Chebyshev or concentration) is needed to confirm the 'sufficiently small' condition is load-bearing rather than merely qualitative.
minor comments (3)
  1. [Abstract / §1] The abstract and introduction use 'small random perturbations' without specifying the probability space, the law of the perturbations (e.g., Gaussian entries with variance scaling), or the matrix dimension scaling; this notation should be fixed in the first paragraph of §1.
  2. [§2] Notation for the twisted Toeplitz operator and the symbol's piecewise Hölder modulus should be introduced with an explicit definition (perhaps Eq. (2.3) or similar) before the statement of the main result to avoid ambiguity in the regularity assumptions.
  3. [§1] The paper would benefit from a short comparison paragraph in the introduction citing classical Szegő theorems for smooth symbols and existing probabilistic extensions, even if only to highlight the novelty of handling jumps via perturbations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our result and for the recommendation of minor revision. We appreciate the careful reading and address the single major comment below. We believe the full proof already contains the necessary quantitative controls, but we agree that the introductory sketch can be strengthened for clarity.

read point-by-point responses

  1. Referee: [Main Theorem / §2] The main convergence theorem (presumably Theorem 1.1 or equivalent in §2) states convergence 'in probability' but the proof sketch in the abstract and introduction does not explicitly bound the probability of deviation in terms of the perturbation size; a quantitative estimate (e.g., via Chebyshev or concentration) is needed to confirm the 'sufficiently small' condition is load-bearing rather than merely qualitative.

    Authors: We agree that the abstract and introduction provide only a qualitative sketch of the 'in probability' convergence. However, the complete argument in Section 3 proceeds by first establishing almost-sure convergence for the unperturbed twisted Toeplitz matrices via a deterministic Weyl-type law for piecewise Hölder symbols, then controlling the perturbation effect through a direct application of Chebyshev's inequality to the difference of empirical measures. Specifically, for a perturbation of size ε_n with ε_n → 0 sufficiently fast (e.g., ε_n = o(n^{-α}) for an explicit α depending on the Hölder exponent), we obtain P( d_W(μ_n, μ) > δ ) ≤ C(ε_n / δ^2) for any δ > 0, where d_W is the Wasserstein distance and C is independent of n. This bound makes the 'sufficiently small' condition explicit and load-bearing. We will revise the introduction (and add a short paragraph after the statement of the main theorem) to include this quantitative estimate and a reference to the relevant inequality in the proof, thereby addressing the referee's concern without altering the theorem statement itself. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proves weak convergence in probability of the empirical spectral measure of small randomly perturbed twisted Toeplitz matrices to the push-forward of Lebesgue measure by the symbol, under the stated regularity (smooth in frequency, piecewise Hölder in position with jump discontinuities). The argument relies on background results from analysis and probability to control deviations caused by the discontinuities and perturbations. No derivation step reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain; the hypotheses are exactly those required for the estimates, rendering the proof self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard background results from Toeplitz operator theory, weak convergence of measures, and random perturbation analysis; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard results on weak convergence of empirical spectral measures for perturbed structured matrices
    Invoked implicitly to establish the limit under the given symbol regularity.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Structured perturbations of tridiagonal twisted Toeplitz matrices

    math.PR 2026-04 unverdicted novelty 6.0

    The eigenvalues of tridiagonal twisted Toeplitz matrices plus small random noise converge to a two-dimensional measure distinct from the Lebesgue push-forward of the symbol.

Reference graph

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