Speaker
Description
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 talk, I will discuss a pertinent counterexample in the context of the generalized Lotka-Volterra equations. Using dynamic mean-field theory, I derive the statistics of the interactions between species in a persistent ecological community. I will 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, we see that the universality principle fails in this system. I thus argue 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.
