Very persistent random walkers reveal transitions in landscape topology

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.

Figures (6)

• Figure 1: The behavior of a passive (black) and persistent (white) random walker on level sets of a cartoon landscape. The persistent walker remains ergodic at lower energies than the passive one. Increasing persistence pushes the ergodic transition of the walk towards a topological transition of the landscape associated with the typical connectivity of configurations.
• Figure 2: Schematic depiction of the energy level set topology of spherical spin glasses as energy density is varied. The threshold $E_\text{th}$ is defined by the energy at which minima begin to outnumber saddle points. In pure models the level set is connected above $E_\text{th}$ while no connected component exists below $E_\text{th}$. In mixed models the breaking up of the level set occurs over a range of energies. $E_\text{alg}$ is the lowest energy at which a large connected component of the level set exists Gamarnik_2021-10_The, while $E_\text{sh}$ marks where the Euler characteristic of the level set changes sign Kent-Dobias_2025_On. In pure models, $E_\text{th}=E_\text{alg}=E_\text{sh}$. An aim of this manuscript is to determine the lowest energy at which typical parts of the level set belong to a large connected component.
• Figure 3: The ergodicity-breaking transition energy as a function of persistence time $\tau_0$ of an active random walker on the microcanonical configuration space of several spherical spin glasses. The solid black line is the explicit solution \ref{['eq:ed.2-spin']} for the pure 2-spin model, while the points show numeric estimates for several other models.
• Figure 4: The correlation function for the pure 3-spin spherical spin glass at several values of the persistence time $\tau_0$. The plateau overlap $q_\text{d}$ of the equilibrium dynamical transition of $\tau_0=0$ is marked. As $\tau_0$ increases, the plateau of the ergodicity-breaking transition moves to higher overlaps. The energies associated in order of increasing $\tau_0$ are $-0.8157$, $-1.0230$, $-1.1076$, $-1.1421$, $-1.1517$, and $-1.1538$, whereas $E_\text{d}\simeq-0.8165$ and $E_\text{th}\simeq-1.1547$
• Figure 5: Energy difference above the threshold energy as a function of $\beta\mu_0$, where $\mu_0$ is the value of $\mu$ associated with a free persistent random walk given by \ref{['eq:μ₀']}. Each line shows the behavior at fixed persistence time for $\tau_0=2^{-8},2^{-7},\ldots,2^{13}$, with larger $\tau_0$ corresponding to lower endpoints. The dashed black line is an inverse-square power law $\beta^{-2}$. In the $3+5$-spin model, a horizontal line shows the location of $E_\text{sh}$ defined in Ref. Kent-Dobias_2025_On where another aspect of the landscape topology undergoes a transition.