Optimal Persistence Reveals Hidden Topology in Complex Energy Landscapes

TL;DR

This paper demonstrates that optimal persistence analysis uncovers hidden topological features in complex energy landscapes, revealing universal principles applicable to disordered systems.

Contribution

It introduces a method to identify topological transitions in energy landscapes using optimal persistence, applicable across different temperatures and system sizes.

Findings

Peak canyon-finding rate at optimal persistence varies with system size and temperature.

Optimal persistence reveals landscape topology consistent with theoretical predictions.

Temperature dependence enters primarily through thermal velocity, simplifying analysis.

Abstract

Infinite persistence marks the topological transition. For finite persistence, the canyon-finding rate Gamma(tau_p) on the p=2 spherical spin glass forms an inverted-U profile, peaking at an optimal tau_p^*. At low temperature (T=0.05), tau_p^* drops from 10 to 5 as N increases through 128, marking the discrete-to-quasi-continuous GOE crossover. For N=1024, the peak is flat between tau_p=5 and 6 within statistical uncertainties, preventing a more precise determination. For N>=128, the canyon width saturates at xi_eff=1, consistent with the measured tau_p^*=5 when beta=0.4. At higher temperatures (T>=0.15), tau_p^*=10 and beta(T) scales as 1/T, with temperature dependence entering only through v_th = sqrt(2T). For T=0.10 and N>=128, high-resolution scans give tau_p^*=8.0; for N<=64 at the same temperature, coarse scans place tau_p^* in the range 8-10. Thus, optimal persistence reveals the hidden topology of the landscape-a principle expected to be generic in disordered landscapes with entropic bottlenecks.