27 pages, 7 figures, The eigenvalue spectrum of a random matrix often only depends on the first and second moments of its elements, but not on the specific distribution from which they are drawn. The validity of this universality principle is often assumed without proof in applications. In this Letter, we offer a pertinent counterexample in the context of the generalized Lotka-Volterra equations. Using dynamic mean-field theory, we derive the statistics of the interactions between species in an evolved ecological community. We then show that the full statistics of these interactions, beyond those of a Gaussian ensemble, are required to correctly predict the eigenvalue spectrum and therefore stability. Consequently, the universality principle fails in this system. We thus show that the eigenvalue spectra of random matrices can be used to deduce the stability of "feasible" ecological communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account., We acknowledge partial financial support from the Agencia Estatal de Investigación (AEI, MCI, Spain) and Fondo Europeo de Desarrollo Regional (FEDER, UE), under Project PACSS (No. RTI2018-093732-B-C21) and the Maria de Maeztu Program for units of Excellence in R&D, Grant No. MDM-2017-0711 funded by MCIN/AEI/10.13039/501100011033., Peer reviewed

The supplemental file provides further details about the calculation of the eigenvalue spectrum of the reduced interaction matrix. There is also some background information on the dynamic mean-field theory analysis of the Lotka-Volterra equations for completeness. The supplement also contains a discussion of the "bottom-up" construction of the reduced interaction matrices., Peer reviewed