Quenched complexity of equilibria for asymmetric generalized Lotka-Volterra equations

We consider the generalized Lotka-Volterra system of equations with all-to-all, random asymmetric interactions describing high-dimensional, very diverse and well-mixed ecosystems. We analyse the multiple equilibria phase of the model and compute its quenched complexity, i.e. the expected value of the logarithm of the number of equilibria of the dynamical equations. We discuss the resulting distribution of equilibria as a function of their diversity, stability and average abundance. We obtain the quenched complexity by means of the replicated Kac-Rice formalism, and compare the results with the same quantity obtained within the annealed approximation, as well as with the results of the cavity calculation and, in the limit of symmetric interactions, of standard methods to compute the complexity developed in the context of glasses.