The method of moments is a classical technique for showing weak convergence and follows a simple recipe: for any natural number m, compute the mth moments of the random variables of interest, and prove they tend to the mth moment of the claimed limit (this works for some limiting laws, including Gaussian). This approach has been prolific for universality results: for the largest eigenvalues of certain random matrices, justify their asymptotic behavior depends solely on a random variable, show the moments of the latter depend (asymptotically) on few moments of the former, and use the Gaussian case to deduce the limiting behavior. Although Gaussianity can be relaxed considerably, some constraints are indispensable: consider a real-valued Wigner matrix with i.i.d. entries. When the fourth moment of the entry distributions is infinite (heavy-tailed), the largest eigenvalues are known to converge to Poisson point processes, whereas when it is finite (light-tailed), the limits are the same as for Gaussian orthogonal ensembles.

This talk focuses on a subfamily of edge cases, distributions at the boundary between heavy- and light- tailed regimes, and presents a new application of the method of moments, one that allows to obtain the asymptotics of the largest eigenvalues directly, without any comparison to the Gaussian case. A byproduct of this result is a connection between the aforementioned subfamily and two other families, finite-rank perturbations of Wigner matrices and sparse random matrices.

This presentation is based on https://arxiv.org/pdf/2203.08712.pdf.