Topological transition versus algorithmic thresholds when $E_\text{th}$ loses dynamical significance

Ascertain whether, in mean-field models where the threshold energy $E_\text{th}$ lies below proven lower bounds for energies achievable by polynomial-time algorithms (and thus loses its dynamical significance), the microcanonical topological transition identified via infinitely persistent random walkers likewise departs from $E_\text{th}$, revealing an analogous discrepancy in the landscape topology measure developed in this work.

Background

The paper argues that in the models studied, the ergodicity-breaking transition at infinite persistence coincides with the threshold energy $E_\text{th}$ where minima outnumber saddle points, indicating a topological disconnection of typical configurations.

However, in some mean-field spin glass models, rigorous algorithmic results show that $E_\text{th}$ can be lower than energies achievable by any polynomial-time dynamics, implying $E_\text{th}$ cannot universally predict dynamical behavior. The authors raise the question of whether this departure from $E_\text{th}$ also manifests in the topology-based measure introduced here.