Concentration of Equilibria and Relative Instability in Disordered Non-Relaxational Dynamics

Understand

We consider a system of random autonomous ODEs introduced by Cugliandolo et al.

• [22], which serves as a non-relaxational analog of the gradient flow for the spherical p-spin model.
• The asymptotics for the expected number of equilibria in this model was recently computed by Fyodorov [32] in the high-dimensional limit, followed a similar computation for the expected number of stable equilibria by Garcia [38].
• We show that for $p > 9$ the number of equilibria, as well as the number of stable equilibria, concentrate around their respective averages, generalizing recent results of Subag and Zeitouni [61, 64] in the relaxational case.