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arxiv: 2605.16160 · v1 · pith:ZHCXY6XInew · submitted 2026-05-15 · 🧮 math.PR

Revisiting Toeplitz and Hankel random matrices via *-convergence of circulant-type matrices

Arup Bose , Pradeep Vishwakarma This is my paper

Pith reviewed 2026-05-19 18:54 UTC · model grok-4.3

classification 🧮 math.PR MSC 60B20
keywords Toeplitz matrixHankel matrix*-convergencecirculant matrixempirical spectral distributionrandom matricesnon-commutative probability
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The pith

Random symmetric Toeplitz matrices converge in *-distribution to the sum of two non-commuting self-adjoint real Gaussian variables, while Hankel matrices converge to sums of symmetrized Rayleigh variables.

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

The paper first proves the joint *-convergence of a random circulant matrix with a deterministic diagonal matrix. It then transfers this result through explicit algebraic relations that express symmetric Toeplitz matrices as combinations of circulant and reverse-circulant forms, yielding the stated limit. The same technique produces a parallel result for Hankel matrices and recovers the known weak convergence of their empirical spectral distributions as a corollary. This route also supplies an alternative expression for the moments of those limiting distributions. A reader would value the clarification because the limits were previously known to exist but lacked explicit non-commutative descriptions.

Core claim

Exploiting the algebraic connections among circulant, reverse-circulant, and left skew-circulant matrices with Toeplitz and Hankel forms, the symmetric Toeplitz random matrix converges in *-distribution to the sum of two non-commutative self-adjoint variables each distributed as a real Gaussian; the non-symmetric Toeplitz case yields a similar sum with complex Gaussians; the Hankel matrix converges to a sum of two self-adjoint variables each having symmetrized Rayleigh distribution. The empirical spectral distributions of the skew-circulant and left skew-circulant matrices converge almost surely to complex Gaussian and symmetrized Rayleigh laws, respectively.

What carries the argument

Joint *-convergence of a random circulant matrix and a deterministic diagonal matrix, transferred via the linear relations that link circulant-type matrices to Toeplitz and Hankel matrices.

If this is right

  • The empirical spectral distribution of symmetric Toeplitz matrices converges weakly almost surely to the distribution of the sum of the two Gaussian variables.
  • The empirical spectral distribution of Hankel matrices converges to the distribution of the sum of the two symmetrized Rayleigh variables.
  • Moments of the limiting spectral distributions admit an explicit non-commutative expansion that differs from previous expressions.
  • The same technique recovers the almost-sure weak convergence of the spectra for skew-circulant and left skew-circulant matrices.

Where Pith is reading between the lines

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

  • The non-commutative description may allow computation of joint eigenvalue statistics or fluctuation results that are harder to obtain from classical moment methods alone.
  • Similar circulant-based reductions could be tested on other patterned matrices such as persymmetric or Toeplitz-plus-Hankel forms.
  • The approach suggests that many structured random matrices possess limits that decompose into a small number of independent non-commutative elements whose distributions are classical.
  • Numerical simulation of finite Toeplitz matrices against the predicted non-commutative law could provide quick consistency checks before analytic moment comparisons.

Load-bearing premise

The algebraic identities expressing Toeplitz and Hankel matrices in terms of circulant, reverse-circulant, and skew-circulant matrices preserve the *-convergence property when the latter are known to converge.

What would settle it

A direct calculation of the fourth moment of the limiting empirical spectral distribution for the symmetric Toeplitz matrix that fails to match the fourth moment obtained by expanding the sum of the two non-commuting real Gaussian variables.

read the original abstract

We establish the joint $*$-convergence of a random circulant matrix and a specific deterministic diagonal matrix. We also show that the empirical spectral distributions of skew-circulant and left skew-circulant random matrices converge weakly a.s.~to complex Gaussian and symmetrized Rayleigh distributions, respectively. The $*$-convergence of symmetric Toeplitz and Hankel random matrices is well known. So is the weak convergence of their random spectrum. However, not much is known about the limits. We exploit the connections of circulant, reverse circulant, and left skew-circulants with the Hankel and Toeplitz matrices, to show the $*$-convergence of the random symmetric Toeplitz matrix to the sum of two non-commutative self-adjoint variables, each having a real Gaussian distribution. A similar result holds for the non-symmetric Toeplitz matrix, but the variables are not self-adjoint and have a complex Gaussian distribution. The random Hankel matrix is shown to converge in $*$-distribution to a sum of two self-adjoint variables, each of which has a symmetrized Rayleigh distribution. As a consequence of these results, we also obtain a different proof of the convergence of the empirical spectral distribution of symmetric Toeplitz and Hankel matrices, and a slightly different way of expressing the moments of the limit spectral distribution.

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

2 major / 3 minor

Summary. The paper establishes the joint *-convergence of a random circulant matrix and a deterministic diagonal matrix. It proves that the empirical spectral distributions of skew-circulant and left skew-circulant random matrices converge almost surely to the complex Gaussian and symmetrized Rayleigh distributions, respectively. Exploiting algebraic identities linking circulant, reverse-circulant, and left skew-circulant matrices to Toeplitz and Hankel forms, the authors derive the *-convergence of symmetric Toeplitz matrices to the sum of two non-commutative self-adjoint variables each with real Gaussian distribution, a parallel result for non-symmetric Toeplitz matrices with complex Gaussians, and the convergence of Hankel matrices to the sum of two self-adjoint variables each with symmetrized Rayleigh distribution. As corollaries, they obtain an alternative proof of the almost-sure convergence of the empirical spectral distributions for symmetric Toeplitz and Hankel matrices together with explicit moment expressions for the limiting laws.

Significance. If the derivations hold, the manuscript supplies explicit non-commutative limiting descriptions for the joint laws of Toeplitz and Hankel random matrices that are consistent with known marginal ESD results. The circulant-based route yields a different proof of ESD convergence and concrete moment formulae, which may simplify further calculations in free probability. The joint *-convergence statement for circulant plus diagonal matrices is a technical contribution that could be reused for other structured ensembles.

major comments (2)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: the joint *-convergence of the random circulant C_n and the deterministic diagonal D_n is stated to follow from moment calculations, but the passage from the explicit eigenvalue formula (via Fourier diagonalization) to the non-commutative limit requires verifying that all mixed moments factor according to the free Gaussian law; the current sketch does not display the combinatorial cancellation that rules out non-Gaussian contributions.
  2. [§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the algebraic identity expressing the symmetric Toeplitz matrix as (C + R)/2 is exact only after a deterministic shift of indices; the paper must confirm that this shift does not affect the *-limit, since the shift is a rank-one perturbation whose norm is O(1/sqrt(n)) but whose effect on higher moments needs explicit control.
minor comments (3)
  1. [§2.2] The definition of left skew-circulant matrices in §2.2 uses an indexing convention that differs from some standard references; adding a short comparison table would improve readability.
  2. [§5] In the statement of the main results for Hankel matrices, the two limiting variables are described as 'self-adjoint' but their joint distribution is only characterized marginally; a remark clarifying whether they are free or merely independent in the non-commutative sense would be helpful.
  3. [§6] Several moment formulae in §6 are given without reference to the corresponding free-probability cumulants; a brief sentence linking them to the R-transform would make the expressions more transparent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and recommendation of minor revision. The comments have prompted us to clarify and strengthen the proofs. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the joint *-convergence of the random circulant C_n and the deterministic diagonal D_n is stated to follow from moment calculations, but the passage from the explicit eigenvalue formula (via Fourier diagonalization) to the non-commutative limit requires verifying that all mixed moments factor according to the free Gaussian law; the current sketch does not display the combinatorial cancellation that rules out non-Gaussian contributions.

    Authors: We agree that the original sketch in the proof of Theorem 3.2 was too brief on the mixed moments. In the revised manuscript we have expanded the argument: starting from the explicit Fourier eigenvalues of the circulant, we compute the joint moments with the diagonal entries via a combinatorial expansion over pairings. The calculation shows that all non-Gaussian contributions cancel by the same Wick-type rules that characterize the free Gaussian, with the diagonal acting as a deterministic multiplier that preserves freeness. The new version includes the low-order moment verifications and the general cancellation argument. revision: yes

  2. Referee: [§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the algebraic identity expressing the symmetric Toeplitz matrix as (C + R)/2 is exact only after a deterministic shift of indices; the paper must confirm that this shift does not affect the *-limit, since the shift is a rank-one perturbation whose norm is O(1/sqrt(n)) but whose effect on higher moments needs explicit control.

    Authors: We concur that the deterministic index shift must be controlled. The shift produces a rank-one perturbation whose operator norm is O(1/sqrt(n)). We have inserted a short lemma after Equation (4.3) that bounds the difference in all joint moments: the contribution of the perturbation to any fixed moment is O(1/sqrt(n)) uniformly in the test polynomials, which tends to zero. Consequently the *-limit is unaffected. The argument uses only the uniform boundedness of the original matrices and the vanishing norm of the perturbation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via new joint convergence result

full rationale

The paper directly establishes the joint *-convergence of a random circulant matrix and a deterministic diagonal matrix as a core new result. It then applies exact algebraic identities (such as Toeplitz = (C + R)/2 and Hankel = (C - R)/2 up to indexing) to transfer this to the claimed *-limits for Toeplitz and Hankel matrices, yielding sums of independent non-commutative Gaussians or symmetrized Rayleigh variables. Marginal limits for skew-circulant matrices follow from Fourier diagonalization plus CLT on eigenvalues. These steps supply independent content and an alternative proof of ESD convergence without any reduction of the target limits to fitted inputs, self-definitional loops, or load-bearing self-citations by construction. The derivation remains externally verifiable through the explicit moment calculations and standard matrix relations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard random-matrix assumptions and the non-commutative probability framework; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Matrix entries are i.i.d. with zero mean and unit variance (or finite moments sufficient for the convergence).
    This is the usual hypothesis under which almost-sure weak convergence of empirical spectral distributions is proved for structured random matrices.

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Reference graph

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