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Title: Relative Instability and Concentration of Equilibria in Non-Gradient Dynamics.

Abstract. A classical picture in the theory of complex high-dimensional random functions is that the gradient dynamics of the function may become slowed and trapped by an abundance of saddles and sinks in the energy landscape. Specifically, when a model has an exponentially large number of such points, the dynamics are expected to be slow and "glassy", a belief now confirmed in a variety of models.

In non-gradient dynamics however, another phase is possible. In particular, work of Ben Arous, Fyodorov, and Khorozhenko on the generalized May-Wigner model found evidence that if the strength of the non-gradient terms were brought past a certain threshold, the total number of equilibria (stationary points) would remain exponentially large, while the number of stable equilibria would go from being exponentially large, to simply vanishing entirely. These two regimes were coined as relative and absolute instability, respectively, and have since been predicted to occur in a variety of models.

We rigorously confirm this picture in the non-gradient analog for the spherical p-spin model.

To do so, we demonstrate concentration of the "annealed" complexity (exponential order) of stable and general equilibria, recently computed by Fyodorov and Garcia, around their typical values, mirroring recent work of Subag and Zeitouni in the relaxational case. The key input is a new computation for the higher moments of the characteristic polynomials of matrices sampled from the elliptic ensemble, as well as general extensions of the recent framework of Ben Arous, Bourgade, and McKenna to the non-relaxational case.

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