A classical picture in the theory of complex high-dimensional random functions is that an exponentially large number of critical points causes the gradient dynamics of the function to become slow and "glassy", becoming trapped in local minima. In non-gradient dynamics however, another case is possible. Here, one may have an exponentially large number of equilibria. but have none that are stable, leading to an endless cycle of wandering around saddles. This is believed to occur when the strength of the non-gradient terms is brought past a certain point, a phenomenon coined Ben Arous, Fyodorov, and Khoruzhenko as the relative-absolute instability transition, and since predicted to occur in a variety of models.
 
We confirm such a transition occurs in the case of the asymmetric p-spin model, the first such rigorous confirmation of the existence of this transition in any model. To do so, we demonstrate concentration of the quenched complexity of stable and general equilibria around their annealed values. Our methods rely on generalizing the recent framework of Ben Arous, Bourgade, and McKenna on the Kac-Rice formula to the non-relaxational case, as well as a computation of moments of the characteristic polynomial of the elliptic ensemble.