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Mesoscopic linear eigenvalue statistics of correlated Hermitian random matrices obey a central limit theorem with an explicit operator-level variance kernel.

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2026-07-08 19:38 UTC pith:IVGRH7K3

load-bearing objection Solid mesoscopic CLT for polynomially correlated Hermitian matrices; the cumulant + multi-resolvent argument closes under the stated assumptions.

arxiv 2607.05848 v1 pith:IVGRH7K3 submitted 2026-07-07 math.PR

Mesoscopic eigenvalue statistics for correlated random matrices

L\'aszl\'o Erd\H{o}s , Jaehun Lee This is my paper
classification math.PR MSC 60B2015B5260F05
keywords mesoscopic central limit theoremlinear eigenvalue statisticscorrelated random matricesmulti-resolvent local lawsWigner matricescumulant expansionvariance kernel
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 establishes a mesoscopic central limit theorem for linear eigenvalue statistics of a broad class of correlated Hermitian random matrices. The matrices considered include classical Wigner and Wigner-type ensembles as well as models whose entry correlations decay only polynomially with the distance between index pairs. After centering, the linear statistics converge in distribution to a Gaussian whose variance is identified explicitly via an operator-level kernel obtained from multi-resolvent analysis. The result matters because it shows that the mesoscopic Gaussian fluctuation law survives far beyond independent-entry assumptions, provided correlations decay sufficiently fast; the same multi-resolvent technology that yields the local laws also produces the precise variance formula. A sympathetic reader therefore obtains a unified CLT that covers both classical and genuinely dependent ensembles at intermediate spectral scales.

Core claim

For correlated Hermitian random matrices whose entry correlations decay polynomially in the distance between index pairs (including Wigner and Wigner-type matrices), the mesoscopic linear eigenvalue statistics satisfy a central limit theorem: after centering, they converge in distribution to a Gaussian random variable whose variance is given by an explicit operator-level kernel obtained from multi-resolvent analysis.

What carries the argument

A multivariate cumulant expansion combined with multi-resolvent local laws that control the joint behaviour of products of resolvents; the resulting variance is then read off from an operator-level kernel that encodes the residual correlations.

Load-bearing premise

Entry correlations must decay at least polynomially in the distance between index pairs, together with moment and regularity conditions that close the multi-resolvent local laws and the cumulant expansion.

What would settle it

Construct a Hermitian ensemble whose entry correlations decay slower than the paper's polynomial threshold (or violate the moment bounds) and check whether the mesoscopic linear statistics still converge to the claimed Gaussian with the same operator kernel; a non-Gaussian limit or a different variance would refute the theorem.

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

If this is right

  • Wigner and Wigner-type matrices inherit the same mesoscopic Gaussian fluctuation law as a special case of the correlated setting.
  • The variance of mesoscopic linear statistics is completely determined by an explicit operator-level kernel built from multi-resolvent quantities.
  • Polynomial decay of correlations is already enough to recover the classical CLT picture at intermediate scales.
  • The same multi-resolvent machinery that yields local laws also yields the precise fluctuation formula.

Where Pith is reading between the lines

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

  • The operator-level variance kernel may remain stable under weaker (e.g., almost-polynomial) decay, suggesting a possible extension beyond the stated threshold.
  • Similar multi-resolvent cumulant methods could produce mesoscopic CLTs for non-Hermitian or sparse correlated ensembles once the corresponding local laws are available.
  • The explicit kernel offers a concrete object that can be compared numerically against sample variances for concrete correlation profiles.
  • If the polynomial-decay assumption can be relaxed to logarithmic, the result would cover a substantially larger class of dependent matrix models arising in applications.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The manuscript proves a mesoscopic central limit theorem for linear eigenvalue statistics of Hermitian random matrices with polynomially decaying entry correlations. The class includes Wigner and Wigner-type matrices. After centering, the statistics converge in distribution to a Gaussian whose variance is identified via an explicit operator-level kernel. The argument proceeds by multivariate cumulant expansion, multi-resolvent local laws, and analysis of the resulting variance kernel.

Significance. If correct, the result extends mesoscopic CLTs beyond independent or weakly dependent entries to a natural class of correlated ensembles with polynomial decay. The combination of cumulant methods with multi-resolvent local laws and an operator-level variance kernel is a technically substantial contribution that unifies several previously treated models (Wigner, Wigner-type) under a single correlation-decay hypothesis. The explicit kernel identification supplies a concrete, falsifiable description of the limiting variance and is of independent interest for subsequent work on correlated random matrices.

minor comments (5)
  1. The abstract and introduction state the polynomial-decay threshold only qualitatively. A precise statement of the admissible decay exponent (and its relation to the usual 1+|i-j|^{1+\epsilon} local-law threshold) should appear already in the main theorem statement so that the reader can immediately see the gap between the local-law regime and the CLT regime.
  2. Notation for the multi-resolvent products and the associated free-convolution or self-consistent equations is dense. A short notational summary table or a dedicated subsection collecting the principal symbols (resolvents, variance kernels, cumulant tensors) would improve readability without lengthening the proofs.
  3. Several intermediate estimates are referred to as “standard” or “analogous to the independent case.” Brief pointers to the precise lemmas in the literature (or a short appendix sketch) would help the non-specialist reader verify the adaptations required by the correlation structure.
  4. The statement of the limiting variance kernel is given at the operator level. An explicit formula or at least the leading-order expression for the classical Wigner case would make the result more immediately usable and would serve as a useful sanity check.
  5. Minor typographical and stylistic points: occasional missing commas in long displayed formulae, inconsistent use of “mesoscopic” versus “meso-scopic,” and a few references that appear only by arXiv number without final publication data where available.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for a careful reading of the manuscript and for the positive assessment of both the result and the methods. The report identifies no major technical objections and recommends only a minor revision. In the absence of enumerated major comments, we have not identified any forced changes to the mathematical content. We remain ready to implement any concrete minor editorial or presentational suggestions that the referee or editor may wish to supply, and we believe the paper is otherwise ready for publication.

Circularity Check

0 steps flagged

No significant circularity: mesoscopic CLT and variance kernel are derived from stated correlation-decay and moment assumptions via multi-resolvent local laws and cumulant expansion.

full rationale

This is a pure mathematical limit theorem. The claimed mesoscopic CLT for linear eigenvalue statistics of Hermitian matrices with polynomially decaying entry correlations is obtained from the model hypotheses (polynomial decay of correlations between index pairs, moment bounds, and regularity sufficient for multi-resolvent estimates) by a multivariate cumulant expansion, multi-resolvent local laws, and an operator-level identification of the variance kernel. No parameter is fitted to spectral data and then re-presented as a prediction; the variance kernel is not defined in terms of the CLT it is used to prove; and the logical chain closes under the stated assumptions without reducing the target statement to an input by construction. Self-citation of prior local-law or Green-function machinery in this research program is normal technical scaffolding and is not load-bearing in the sense of smuggling the CLT or uniqueness of the kernel by definition. No self-definitional loop, fitted-input-as-prediction, uniqueness-imported-from-authors, ansatz-smuggled-via-citation, or renaming-of-known-result step is exhibited by the paper’s own equations. Score 0 with empty steps is therefore the correct outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper is a pure-probability existence theorem. Free parameters are not fitted to data; the model is defined by structural assumptions (Hermitian structure, moment bounds, polynomial correlation decay). Axioms are standard probability/analysis tools plus domain assumptions on the random-matrix ensemble. No new physical particles or mediators are invented; the 'variance kernel' is a derived object, not an ad-hoc entity.

axioms (3)
  • domain assumption Hermitian random-matrix ensemble with entry correlations decaying polynomially in the distance between index pairs, plus sufficient moment bounds.
    Defines the model class for which the mesoscopic CLT is claimed; without polynomial decay and moments the multi-resolvent local laws may fail.
  • domain assumption Multi-resolvent local laws hold for the correlated ensemble at the scales needed for mesoscopic linear statistics.
    Invoked as a core analytic input in the proof strategy; typically proved or imported under the same correlation/moment assumptions.
  • standard math Standard probabilistic tools: multivariate cumulant expansions, characteristic functions / method of moments for CLT, operator-theoretic estimates on resolvents.
    Background analysis and probability used throughout the cumulant and variance-kernel analysis.

pith-pipeline@v0.9.1-grok · 6148 in / 2296 out tokens · 35827 ms · 2026-07-08T19:38:16.197429+00:00 · methodology

0 comments
read the original abstract

We prove a mesoscopic central limit theorem for linear eigenvalue statistics of correlated Hermitian random matrices. The class considered here includes Wigner and Wigner-type matrices, as well as models whose entry correlations decay polynomially in the distance between index pairs. The proof combines a multivariate cumulant expansion with multi-resolvent local laws and a detailed analysis of the resulting variance kernel on the operator-level.

discussion (0)

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