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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 several limiting laws, including Gaussian, but not all). This approach has been prolific for universality results: for the largest eigenvalues of 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, some constraints are indispensable, and these are oftentimes related to fourth moments: the largest eigenvalues of Wigner matrices whose entries have heavy- tailed distributions (roughly speaking, these are laws with infinite fourth moments) are known to converge to Poisson point processes, whereas for light-tailed, the limits are the same as for Gaussian ensembles (Tracy-Widom distributions). 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 asymptotic limits of the largest eigenvalues directly, without any comparison to the Gaussian case (this is based on the combinatorial device developed by Sinai and Soshnikov in 1998). A byproduct of this 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.