Very persistent random walkers reveal transitions in landscape topology

TL;DR

This paper investigates how persistent random walks behave on complex energy landscapes, revealing a link between ergodicity-breaking transitions and topological changes in the configuration space, with implications for understanding glassy systems.

Contribution

It demonstrates that in certain models, the ergodicity-breaking transition aligns with a topological transition in the configuration space, especially in the limit of infinite persistence time.

Findings

Persistent walks remain ergodic below the glass transition energy.

In models with well-understood landscapes, the transition coincides with a topological change.

The topological transition energy can be identified even when landscape details are unclear.

Abstract

We study the typical behavior of random walkers on the microcanonical configuration space of mean-field disordered systems. Passive walks have an ergodicity-breaking transition at precisely the energy density associated with the dynamical glass transition, but persistent walks remain ergodic at lower energies. In models where the energy landscape is thoroughly understood, we show that, in the limit of infinite persistence time, the ergodicity-breaking transition coincides with a transition in the topology of microcanonical configuration space. We conjecture that this correspondence generalizes to other models, and use it to determine the topological transition energy in situations where the landscape properties are ambiguous.