Covariance kernel of linear spectral statistics for half-heavy tailed wigner matrices

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 clased 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 [F. Benaych-Georges and A. Maltsev, Fluctuations of linear statistics of half-heavy-tailed random matrices, Stochastic Process. Appl. 126(11) (2016) 3331 3352]. 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.