Typical number and stability of equilibria in large non-reciprocal complex systems

Abstract

We consider a prototypical class of dynamics of $N\gg 1$ degrees of freedom subject to two competing terms, one favoring relaxation to a trivial equilibrium and one taking the form of a Gaussian force field promoting the emergence of many local equilibria. The latter is a non-gradient field, i.e. it includes a non-reciprocal component with a tunable strength. For this model, we compute the quenched complexity, i.e. the exponential growth rate of the typical number of equilibria, resolved by stability index. We compare it with the annealed complexity, i.e. the exponential growth rate of the average number of equilibria. We show that the two coincide only in the fully non-reciprocal limit: for any finite degree of reciprocity, typical realizations contain exponentially fewer equilibria than predicted by the average count. Increasing reciprocity enhances the discrepancy between the two. The quenched complexity depends explicitly on the covariance of the random force field and on the degree of non-reciprocity: the universality observed at the annealed level breaks down. The average counting of equilibria correctly identifies the location of the topology trivialization transition, but it does not capture the typical organization of equilibria near this transition, nor the stability index of the dominant equilibria.