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

TL;DR

This paper analyzes a high-dimensional disordered dynamical system, showing that the number of equilibria concentrates around the average for p > 9 and identifying a transition from relative to absolute instability.

Contribution

It extends the understanding of equilibrium concentration and instability transitions to non-relaxational dynamics in high-dimensional random systems.

Findings

Number of equilibria concentrates for p > 9

Transition from relative to absolute instability confirmed

Generalizes relaxational case results to non-relaxational dynamics

Abstract

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. In particular, we confirm that this model undergoes a transition from relative to absolute instability, in the sense of Ben Arous, Fyodorov, and Khoruzhenko [11].