Covariance Kernel of Linear Spectral Statistics for Half-Heavy tailed Wigner Matrices

TL;DR

This paper derives an explicit covariance kernel for the Gaussian process describing fluctuations of linear spectral statistics in Wigner matrices with half-heavy tails, revealing how large eigenvalues influence spectral fluctuations.

Contribution

It provides a closed-form, alpha-dependent covariance kernel for the spectral fluctuations of Wigner matrices with limited moments, extending understanding of spectral behavior in heavy-tailed regimes.

Findings

Explicit alpha-dependent covariance kernel derived

Integral kernel representation of covariance obtained

Heuristic interpretation of large eigenvalues' impact provided

Abstract

In this paper we analyze the covariance kernel of the Gaussian process that arises as the limit of fluctuations of linear spectral statistics for Wigner matrices with a few moments. More precisely, the process we study here corresponds to Hermitian matrices with independent entries that have $\alpha$ moments for $2<\alpha < 4$. We obtain a closed form $\alpha$-dependent expression for the covariance of the limiting process resulting from fluctuations of the Stieltjes transform by explicitly integrating the known double Laplace transform integral formula obtained in the literature. We then express the covariance as an integral kernel acting on bounded continuous test functions. The resulting formulation allows us to offer a heuristic interpretation of the impact the typical large eigenvalues of this matrix ensemble have on the covariance structure.