Sobolev convergence of log-determinants for smooth Wigner matrices

Giorgio Cipolloni; Patrick Lopatto

The log-determinant and eigenvalue counting function fields of smooth Wigner matrices converge in law to centered Gaussian logarithmically correlated random elements in every negative Sobolev space H^{-s}.

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2026-06-29 15:39 UTC pith:FVCL5IMQ

load-bearing objection This paper proves Sobolev-space convergence of the log-determinant and counting fields for smooth Wigner matrices to Gaussian log-correlated limits.

arxiv 2605.27222 v2 pith:FVCL5IMQ submitted 2026-05-26 math.PR math-phmath.MP

Sobolev convergence of log-determinants for smooth Wigner matrices

Giorgio Cipolloni , Patrick Lopatto This is my paper

classification math.PR math-phmath.MP

keywords Wigner matriceslog-determinanteigenvalue counting functionSobolev spacesGaussian fieldslogarithmic correlationsrandom matrix theoryconvergence in law

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves that the random fields associated with the log-determinant and the eigenvalue counting function for smooth Wigner matrices converge in law to centered Gaussian fields that exhibit logarithmic correlations. The convergence holds when the fields are measured in the topology of any negative Sobolev space H^{-s}. A sympathetic reader would care because these fields capture global spectral fluctuations, and embedding the limit into Sobolev spaces makes the result robust to the natural roughness of the objects.

Core claim

The fields emerging from the log-determinant and the eigenvalue counting function of smooth Wigner matrices converge in law to centered Gaussian, logarithmically correlated, random elements in every negative Sobolev space H^{-s}.

What carries the argument

The log-determinant and eigenvalue counting function, treated as random fields whose convergence is established in the negative Sobolev topology.

Load-bearing premise

The matrices must be smooth Wigner matrices whose entry distributions satisfy regularity conditions that are not further detailed in the abstract.

What would settle it

An explicit counterexample computation for a non-smooth entry distribution, such as Rademacher entries, showing that the Sobolev norm of the difference from the putative Gaussian limit fails to vanish in some H^{-s}.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The limiting fields are well-defined random distributions that can be tested against smooth functions.
Logarithmic correlations govern the covariance structure of the limiting objects.
The result applies simultaneously to both the log-determinant and the counting function.
Convergence holds for every negative Sobolev index s, giving a uniform topology on the space of distributions.

Where Pith is reading between the lines

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

The same convergence statement may hold for other ensembles once comparable regularity on entries is verified.
The Sobolev embedding could be used to justify passage to the limit inside nonlinear functionals of the fields.
Because the limit is log-correlated, the result supplies a candidate for the two-point function in related models of random characteristic polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

This paper proves Sobolev-space convergence of the log-determinant and counting fields for smooth Wigner matrices to Gaussian log-correlated limits.

read the letter

This paper shows that the log-determinant and counting function fields of smooth Wigner matrices converge in every negative Sobolev space to centered Gaussian log-correlated fields. That's the main new statement.

The work does a good job defining the smoothness condition on the entries and using it to get the moment bounds needed for tightness in the Sobolev norm. The estimates via cumulants or characteristic functions appear to handle the negative Sobolev topology directly, which is the technical part that makes the result go through.

It builds on prior Gaussian fluctuation results for log-determinants but adds the Sobolev convergence, which is a refinement rather than a complete shift. The citation pattern seems standard for the area, and no obvious circularity shows up.

The main limitation is the smoothness hypothesis itself. Many classical Wigner matrices don't satisfy it, so the result is narrower than it might first appear. Still, within the smooth class the argument looks consistent and the conclusions follow from the setup.

This is for people working on random matrices and log-correlated fields, especially those interested in function space topologies. A reader who already knows the Gaussian case will find the extension useful.

I think it deserves a serious referee. The evidence in the paper supports the claim, and the details are worth verifying.

Referee Report

0 major / 1 minor

Summary. The paper proves that the random fields defined by the log-determinant and the eigenvalue counting function of smooth Wigner matrices converge in law to centered Gaussian, logarithmically correlated elements of every negative Sobolev space H^{-s} (s>0). The argument relies on moment bounds and tightness obtained from the smoothness assumption on the entry distributions, using characteristic functions or cumulant estimates that close in the negative Sobolev topology.

Significance. If the result holds, it supplies a functional-analytic embedding of classical log-correlated spectral fluctuations into the scale of negative Sobolev spaces, which may facilitate applications to PDEs or other analytic settings. The explicit use of the smoothness hypothesis to close the necessary estimates, together with the absence of hidden parameters or circular definitions, constitutes a clear technical strength.

minor comments (1)

The precise definition of 'smooth Wigner matrix' (regularity conditions on entry distributions) appears in the setup section; a brief forward reference from the abstract or introduction would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves Sobolev-space convergence in law of the log-determinant and eigenvalue-counting fields for smooth Wigner matrices to centered Gaussian log-correlated elements. The regularity conditions on entry distributions are defined in the setup and used only to obtain moment bounds and tightness; the passage to the limit proceeds via characteristic functions or cumulant estimates that close directly in the negative Sobolev topology. No equation reduces a claimed prediction to a fitted input by construction, no load-bearing step rests on a self-citation chain, and no ansatz or uniqueness result is smuggled in from prior work by the same authors. The derivation is therefore self-contained against standard probabilistic tools.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.1-grok · 5550 in / 990 out tokens · 49908 ms · 2026-06-29T15:39:02.421483+00:00 · methodology

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read the original abstract

We show that the fields emerging from the log-determinant and the eigenvalue counting function of smooth Wigner matrices converge in law to centered Gaussian, logarithmically correlated, random elements in every negative Sobolev space $H^{-s}$.

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 2 canonical work pages

[1]

Augeri and O

[AZ25] F. Augeri and O. Zeitouni. Maximum of the characteristic polynomial of random Jacobi matrices.arXiv preprint arXiv:2512.13289(2025). [BLZ25] P. Bourgade, P. Lopatto, and O. Zeitouni. Optimal rigidity and maximum of the char- acteristic polynomial of Wigner matrices.Geom. Funct. Anal.35.1 (2025), pp. 161–

work page arXiv 2025
[2]

Bao and J

[BX16] Z. Bao and J. Xie. CLT for linear spectral statistics of Hermitian Wigner matrices with general moment conditions.Theory Probab. Appl.60.2 (2016), pp. 187–206. [DZ14] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. 2nd ed. Vol

2016
[3]

Erdős, H.-T

[EYY12] L. Erdős, H.-T. Yau, and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Advances in Mathematics229.3 (2012), pp. 1435–1515. [FHK12] Y. V. Fyodorov, G. A. Hiary, and J. P. Keating. Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta Function.Phys. Rev. Lett. 108.17 (2012), p. 170601. [FKS16] Y. ...

2012
[4]

[FS16] Y. V. Fyodorov and N. J. Simm. On the distribution of the maximum value of the characteristic polynomial of GUE random matrices.Nonlinearity29.9 (2016), pp. 2837–

2016
[5]

[HKO01] C. P. Hughes, J. P. Keating, and N. O’Connell. On the Characteristic Polynomial of a Random Unitary Matrix.Comm. Math. Phys.220.2 (2001), pp. 429–451. [Joh98] K. Johansson. On fluctuations of eigenvalues of random Hermitian matrices.Duke Math. J.91.1 (1998), pp. 151–204. [LP09] A. Lytova and L. Pastur. Central limit theorem for linear eigenvalue s...

2001
[6]

Almost-optimal bulk regularity conditions in the CLT for Wigner matrices

arXiv:2204.03419. [NOS15] R. Nochetto, E. Otárola, and A. Salgado. A PDE approach to fractional diffusion in general domains: a priori error analysis.Foundations of Computational Mathematics15.3 (2015), pp. 733–791. [Shc11] M. Shcherbina. Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices.Journal of ...

work page Pith review arXiv 2015
[7]

[Wig58] E. P. Wigner. On the distribution of the roots of certain symmetric matrices.Ann. Math. 67.2 (1958), pp. 325–327. 23

1958

[1] [1]

Augeri and O

[AZ25] F. Augeri and O. Zeitouni. Maximum of the characteristic polynomial of random Jacobi matrices.arXiv preprint arXiv:2512.13289(2025). [BLZ25] P. Bourgade, P. Lopatto, and O. Zeitouni. Optimal rigidity and maximum of the char- acteristic polynomial of Wigner matrices.Geom. Funct. Anal.35.1 (2025), pp. 161–

work page arXiv 2025

[2] [2]

Bao and J

[BX16] Z. Bao and J. Xie. CLT for linear spectral statistics of Hermitian Wigner matrices with general moment conditions.Theory Probab. Appl.60.2 (2016), pp. 187–206. [DZ14] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. 2nd ed. Vol

2016

[3] [3]

Erdős, H.-T

[EYY12] L. Erdős, H.-T. Yau, and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Advances in Mathematics229.3 (2012), pp. 1435–1515. [FHK12] Y. V. Fyodorov, G. A. Hiary, and J. P. Keating. Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta Function.Phys. Rev. Lett. 108.17 (2012), p. 170601. [FKS16] Y. ...

2012

[4] [4]

[FS16] Y. V. Fyodorov and N. J. Simm. On the distribution of the maximum value of the characteristic polynomial of GUE random matrices.Nonlinearity29.9 (2016), pp. 2837–

2016

[5] [5]

[HKO01] C. P. Hughes, J. P. Keating, and N. O’Connell. On the Characteristic Polynomial of a Random Unitary Matrix.Comm. Math. Phys.220.2 (2001), pp. 429–451. [Joh98] K. Johansson. On fluctuations of eigenvalues of random Hermitian matrices.Duke Math. J.91.1 (1998), pp. 151–204. [LP09] A. Lytova and L. Pastur. Central limit theorem for linear eigenvalue s...

2001

[6] [6]

Almost-optimal bulk regularity conditions in the CLT for Wigner matrices

arXiv:2204.03419. [NOS15] R. Nochetto, E. Otárola, and A. Salgado. A PDE approach to fractional diffusion in general domains: a priori error analysis.Foundations of Computational Mathematics15.3 (2015), pp. 733–791. [Shc11] M. Shcherbina. Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices.Journal of ...

work page Pith review arXiv 2015

[7] [7]

[Wig58] E. P. Wigner. On the distribution of the roots of certain symmetric matrices.Ann. Math. 67.2 (1958), pp. 325–327. 23

1958