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arxiv: 2604.26499 · v1 · submitted 2026-04-29 · 🧮 math.PR · math-ph· math.MP

Spectrum of Random Matrices with Exploding Moments

Indrajit Jana , Sunita Rani This is my paper

Pith reviewed 2026-05-07 11:27 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords central limit theoremrandom matricesexploding momentslinear eigenvalue statisticsWick formulaelliptic matricescirculant matricesblock matrices
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The pith

Central limit theorems hold for linear eigenvalue statistics of random matrices with exploding moments

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

The paper aims to prove central limit theorems for linear eigenvalue statistics in random matrix models where entry moments increase with the matrix size. It covers elliptic, centrosymmetric, circulant, and inter-correlated block matrices, deriving the results via the asymptotic Wick formula. A reader would care if true because this extends the reach of classical random matrix theory to models with growing fluctuations, which appear in applications with non-stationary data or size-dependent noise. The work shows that the explosion rate can be controlled so that the limiting distribution remains Gaussian with explicit variance.

Core claim

For random matrices whose entries have moments that explode with the dimension n, the properly centered and scaled linear eigenvalue statistics converge in distribution to a normal law, with the limiting covariance determined by the asymptotic Wick formula, for each of the four matrix classes studied.

What carries the argument

The asymptotic Wick formula, which provides the limiting joint moments of the linear statistics by summing over pairings that respect the matrix dependence structure.

If this is right

  • The CLT applies directly to elliptic matrices with exploding moments.
  • Centrosymmetric matrices with the same property also satisfy the CLT.
  • Circulant and inter-correlated block matrices follow the same limiting behavior.
  • The variance of the limiting Gaussian is explicitly computable from the formula without further approximation.

Where Pith is reading between the lines

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

  • Numerical simulations on moderate-sized matrices could verify the rate at which the distribution approaches normality.
  • The approach may generalize to other structured matrices like Toeplitz or Hankel with similar moment growth.
  • In applications, this would justify using normal approximations for test statistics derived from such matrices even when variances increase with sample size.

Load-bearing premise

The explosion of moments occurs at a rate compatible with the conditions needed for the asymptotic Wick formula to yield a Gaussian limit, and the dependence patterns in each matrix family satisfy the required moment bounds.

What would settle it

Generate a large number of realizations of one such matrix ensemble, compute the linear statistic for a smooth test function, and test whether the empirical distribution is Gaussian with the variance predicted by the Wick formula; a clear mismatch in the tails or variance would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.26499 by Indrajit Jana, Sunita Rani.

Figure 1
Figure 1. Figure 1: Examples of graphs satisfying the conditions of thick tree (left) and a graph violating them (right). The green-colored edges highlight the violations of the defining conditions. Definition 2.4 (Fat Tree). A directed graph T = (V, E) is called a fat tree if it is a connected graph without loops and all the edges have multiplicity more than one. Additionally, the underlying simple graph T = (V, E), obtained… view at source ↗
Figure 2
Figure 2. Figure 2: Examples of graphs satisfying the conditions of fat tree (top-left) and graphs that do not. The green-colored edges highlight the violations of the defining conditions. where qk,l denote the number of ordered vertex pairs (i, j) with i < j such that there are exactly k edges from vertex i to vertex j and l edges from vertex j to vertex i in thick tree Tπ. (c) For α < 1 we have E view at source ↗
Figure 3
Figure 3. Figure 3: Examples illustrating colored fat trees (panels (a) and (b)) and viola￾tions of the defining conditions (panels (c)–(f)). =  1 √ N Tr(Mk ) − E  1 √ N Tr(Mk )  k≥1 =  1 √ N  Tr(M(1)) k + Tr(M(2)) k  − E  1 √ N  Tr(M(1)) k + Tr(M(2)) k   k≥1 . To prove the convergence of n ZN (k) o k≥1 to a Gaussian process, it is sufficient to prove the conver￾gence of n ZN (Tπ) o π∈P(k) , where (4.4) ZN (Tπ) = √ N  view at source ↗
read the original abstract

We study the central limit theorem (CLT) for linear eigenvalue statistics of several types of matrix models, whose entries are having exploding moments, i.e., moments of the entries are increasing with the size of the matrix. In particular, we study elliptic, centrosymmetric, circulant, and inter-correlated block matrices. The CLTs are established using asymptotic Wick formula.

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 / 1 minor

Summary. The manuscript claims to establish central limit theorems for linear eigenvalue statistics of elliptic, centrosymmetric, circulant, and inter-correlated block random matrices whose entry moments explode (increase) with matrix size n, with the proofs relying on application of the asymptotic Wick formula.

Significance. If the moment-growth conditions are verified to be compatible with the Wick-formula error bounds, the results would extend existing CLT theory for linear statistics to a class of non-stationary or heavy-tailed matrix ensembles, providing a unified treatment across four structured families.

major comments (2)
  1. Abstract: the claim that CLTs are established for all four matrix families rests on the asymptotic Wick formula, yet the text provides no explicit verification that the exploding-moment rates satisfy the o(1) remainder bounds on non-pairing (higher-cumulant) terms for each dependence structure (row/column sums for elliptic, block correlations, circulant periodicity, etc.).
  2. Main derivations (sections applying the Wick formula): without deriving or citing the precise growth threshold (e.g., E[|X_ij|^k] = o(n^{f(k)}) relative to variance and dependence) separately for each model, it is impossible to confirm that the formula applies without additional restrictions on the explosion rate.
minor comments (1)
  1. Abstract: minor grammatical phrasing ('entries are having exploding moments') should be revised for precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the asymptotic Wick formula conditions. We address the major comments below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

  1. Referee: [—] Abstract: the claim that CLTs are established for all four matrix families rests on the asymptotic Wick formula, yet the text provides no explicit verification that the exploding-moment rates satisfy the o(1) remainder bounds on non-pairing (higher-cumulant) terms for each dependence structure (row/column sums for elliptic, block correlations, circulant periodicity, etc.).

    Authors: We agree that the abstract would benefit from a brief indication that the moment-growth assumptions are compatible with the Wick-formula error bounds for each dependence structure. In the revised manuscript we will add a short clarifying sentence in the abstract and a dedicated paragraph in the introduction that outlines how the o(1) remainder conditions on higher-cumulant terms are satisfied under the stated explosion rates for elliptic, centrosymmetric, circulant, and block models. revision: yes

  2. Referee: [—] Main derivations (sections applying the Wick formula): without deriving or citing the precise growth threshold (e.g., E[|X_ij|^k] = o(n^{f(k)}) relative to variance and dependence) separately for each model, it is impossible to confirm that the formula applies without additional restrictions on the explosion rate.

    Authors: The manuscript applies the asymptotic Wick formula under the general moment conditions already stated in each section, but we acknowledge that model-specific growth thresholds are not derived explicitly. In the revision we will insert, for each of the four families, a short calculation (or citation to the relevant lemma in the Wick-formula literature) showing that the assumed explosion rate E[|X_ij|^k] = o(n^{f(k)}) is compatible with the required o(1) bound on non-pairing terms, taking into account the respective variance normalizations and dependence structures. This will be placed immediately before the application of the formula in each section. revision: yes

Circularity Check

0 steps flagged

No circularity: CLTs obtained by applying external asymptotic Wick formula to new matrix families

full rationale

The paper establishes CLTs for linear eigenvalue statistics of elliptic, centrosymmetric, circulant, and inter-correlated block matrices whose entries have exploding moments by invoking the asymptotic Wick formula. The abstract and available text give no equations or self-citations in which a claimed prediction or uniqueness result is defined in terms of the target CLT, nor any fitted parameter that is relabeled as a prediction. The derivation chain therefore remains independent of its own outputs and relies on an external combinatorial tool applied to the stated moment and dependence conditions of the new models.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the central claim rests on the applicability of the asymptotic Wick formula and unspecified growth rates of moments.

pith-pipeline@v0.9.0 · 5343 in / 1018 out tokens · 43785 ms · 2026-05-07T11:27:48.974176+00:00 · methodology

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Reference graph

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