Spectrum of Random Matrices with Exploding Moments

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.

• The paper establishes that Gaussian fluctuations for the normalized trace of matrix powers emerge for the critical regime (α=1) using combinatorial graph techniques.
• It details how structured ensembles—including elliptic, circulant, and block-intercorrelated matrices—exhibit distinct universality transitions based on the explosion rate of moments.
• The work introduces new combinatorial classifications, such as thick trees and fat trees, to map moment contributions in the asymptotic limit.

Summary of "Spectrum of Random Matrices with Exploding Moments" (2604.26499)

Introduction and Problem Formulation

This paper investigates the asymptotic spectral properties and fluctuations of random matrix ensembles with entries possessing diverging ("exploding") moments, focusing on the central limit theorem (CLT) for linear eigenvalue statistics (LES). Unlike the conventional (Wigner-type) setting where matrix entries are assumed to have fixed, finite moments, the authors consider cases where, as the matrix size $N$ grows, higher moments scale as $N^{(m/2)-\alpha}$, where $\alpha>0$. Several prominent structured ensembles are treated: elliptic, centrosymmetric, circulant, and block-intercorrelated matrices.

The analysis centers on the normalized traces of matrix powers, $\frac{1}{N}\operatorname{Tr}(A^k)$, and the fluctuations of linear eigenvalue functionals, especially for monomial test functions. The key technical ingredient is a careful combinatorial classification of index matchings and the associated graph-theoretic structures (notably, thick trees and fat trees) that encode when nontrivial contributions persist in the large-$N$ limit.

Main Results: Limiting Fluctuations and Universality Transitions

Elliptic and Related (Correlated Non-Hermitian) Ensembles

The authors consider non-Hermitian matrices with entry covariances that allow for certain symmetries and off-diagonal correlations, with moments diverging at rate controlled by $\alpha$. For these ensembles, the limiting behavior of the normalized trace of powers is explicitly characterized:

• For $\alpha>1$: The expectation of the normalized trace vanishes as $N\to\infty$ due to overwhelming cancellation—no nontrivial limits exist.
• For $\alpha=1$: Gaussian fluctuations persist. The CLT for normalized traces holds with a variance and normalization explicitly determined by the limiting joint moments of the entries, encoded via products over combinatorial thick tree graphs.
• For $\alpha<1$: The normalization diverges unless the underlying combinatorial structure is maximally constrained; cases with single edges or loops yield vanishing contributions. The residual contributions grow with a rate dictated by $N^{(m/2)-\alpha}$0, where $N^{(m/2)-\alpha}$1 is the number of vertices and $N^{(m/2)-\alpha}$2 counts multi-edges in the associated partition graph.

Gaussian fluctuations are established constructively using an asymptotic Wick formula, proven via a detailed multigraph expansion that tracks all nonnegligible contributions in moment calculations. The covariance structure in the $N^{(m/2)-\alpha}$3 case is described precisely by enumerating all tree-wise combinatorial identifications among multiple traces (i.e., the graphs obtained by "gluing" traces at shared edges).

IID Non-Hermitian Ensembles

The completely uncorrelated case (iid non-Hermitian matrices) is addressed in full parallel. The primary difference is that only unordered fat trees (with all edges of multiplicity at least two, no single loops/edges) survive in the limit; the combinatorics and normalization mirror the elliptic ensemble, but with simplifications due to independence. Here, the limiting Gaussian process has a covariance determined by products over unordered multi-edges.

Block-Intercorrelated Ensembles

A block diagonal model with two (potentially correlated) iid blocks is studied. Inter-block correlations between matching entries can be controlled via a general moment joint scaling. For this setting, the spectrum decomposes into a sum of traces over both blocks; the covariance structure of the limiting Gaussian limit is again encoded via colored fat trees (recording from which block each edge originates).

For explicitly centrosymmetric matrices, the symmetry constraint is shown to induce a block-diagonal structure after appropriate orthogonalization (citing classical results in matrix theory), leading directly to a matching analysis as for correlated block ensembles, but with explicit formulas for the limiting mixed moments.

Circulant Matrices

For circulant random matrices with exploding moments, explicit eigenvalue formulas—derived from the discrete Fourier diagonalization—allow exact enumeration of the moments and covariance. The combinatorial complexity reduces to counting index configurations consistent with cyclic summation conditions. In the $N^{(m/2)-\alpha}$4 regime, the limiting centered and normalized traces converge to independent Gaussian variables with variance $N^{(m/2)-\alpha}$5, i.e., the multi-trace covariance kernel is exactly diagonal: $N^{(m/2)-\alpha}$6. The analysis relies on a detailed degree-of-freedom argument: only complete pairings of trace monomials contribute at leading order, and any multi-linking or incomplete pairing results in a strictly lower, negligible contribution.

The authors conjecture that similar techniques apply to other structured ensembles (reverse circulant, symmetric circulant, Toeplitz, Hankel), as the circulant case exhibits substantial technical differences from the unstructured models.

Technical Innovations

• The extension of the Wick formula to the exploding moment regime in an array of structured ensembles, requiring new combinatorial results about graph classes (thick trees, fat trees, colored fat trees).
• The explicit handling of block-structured (particularly centrosymmetric) ensembles, where spectral decompositions under orthogonal similarity produce block-diagonal matrices with explicit inter-block correlations in the exploded moment setting.
• A systematic characterization of the universality transition: passing from trivial ($N^{(m/2)-\alpha}$7) to nontrivial Gaussian fluctuating ($N^{(m/2)-\alpha}$8) to diverging or nilpotent ($N^{(m/2)-\alpha}$9) regimes depending on the interplay between the graph combinatorics and the rate of moment divergence.

Implications and Outlook

On the theoretical level, this work delineates the precise scaling window for universality of classical CLT-type results for LES in random matrices as the tails of the matrix entry distributions become heavier, with the rate parameter $\alpha>0$0 governing the existence and structure of the limit. The structural classification via graphs suggests a robust organizing principle for the analysis of other, more exotic ensembles with additional invariance or dependency patterns.

From a practical perspective, the results provide analytic formulas for limiting eigenvalue fluctuation statistics in settings where entries are heavy-tailed but not in the domain of attraction of classic stable laws; these might be relevant in signal processing, finance, and applied probability scenarios where data matrices exhibit such moment divergence. The explicit identification of the variance/covariance as a function of the explosion rate and the combinatorial structure provides actionable input for evaluating large-dimensional asymptotics in these contexts.

The methods and results point to natural next questions:

• Extension to polynomial or general test functions in LES (beyond monomials).
• Spectral edge behavior and local statistics in the exploding moment regime.
• More general dependency structures and matrix symmetries (e.g., Toeplitz, Hankel, block-Toeplitz).

Conclusion

The paper provides a comprehensive and rigorous framework for understanding the spectrum and asymptotic Gaussian fluctuations of structured random matrices whose entry moments diverge polynomially with dimension. By mapping the moment combinatorics to explicit graph classes, the authors establish precise universality/threshold results and deliver analytic characterizations of the limiting covariance in a range of sparse, block, and cyclic settings. These results advance the mathematical theory of high-dimensional random matrices beyond the classic light-tailed regime, aligning with recent interest in heavy-tailed random phenomena in both theory and applications.