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

We consider a system of random autonomous ODEs introduced by Cugliandolo et al. (Phys Rev Lett 78: 350–353, 1997), 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 (J Stat Mech Theory Exp 12: 124003–21, 2016) in the high-dimensional limit, followed a similar computation for the expected number of stable equilibria by Garcia (Garcia: On the number of equilibria with a given number of unstable directions. arXiv:1709.04021, 2017). We show that for \(p>9\) $p>9$ , the number of equilibria, as well as the number of stable equilibria, concentrate around their respective averages, generalizing recent results of Subag and Zeitouni (Ann Probab 45: 3385–3450, 2017) and (J Math Phys 62: 123301–15, 2021) in the relaxational case. In particular, we confirm that this model undergoes a transition from relative to absolute instability, in the sense of Ben Arous, Fyodorov, and Khoruzhenko (Proc Natl Acad Sci U.S.A. 118: 2023719118–8 2021).