Generic correspondence between ergodicity breaking and topological transition at infinite persistence

Establish, for mean-field disordered systems such as spherical spin glasses (including mixed p-spin variants), that in the limit of infinite persistence time (τ₀ → ∞) for a random walker constrained to a fixed microcanonical energy level H(x)=EN on the N-sphere, the energy level at which the walker’s dynamics lose ergodicity coincides with a topological transition of the microcanonical configuration space—specifically, the transition where typical configurations cease to belong to the same connected component.

Background

The paper studies random walkers confined to the microcanonical configuration space of spherical spin glass models, comparing passive (Markovian) and persistent (non-Markovian) dynamics. Passive walkers exhibit an ergodicity-breaking transition at the energy corresponding to the canonical dynamical glass transition, driven by entropic barriers rather than topology.

For persistent walkers, numerical and exact results indicate that increasing persistence lowers the ergodicity-breaking energy, and in the limit of infinite persistence the transition aligns with the threshold energy E_th at which minima outnumber saddle points—an indicator of a topological change in the microcanonical configuration space. The authors therefore conjecture that, generically, the ergodicity-breaking transition for infinitely persistent walkers marks the topological disconnection of typical configurations.