Critical divergence law of the zero-frequency correlation at the active ergodicity-breaking transition

Determine the power-law governing the divergence of the zero-frequency component of the correlation function’s Fourier transform, \hat C(0), at the ergodicity-breaking transition of persistent (non-Markovian) random walkers on the microcanonical configuration space of spherical spin glass models, thereby enabling precise estimation of the asymptotic transition energy beyond equilibrium conditions.

Background

To locate the ergodicity-breaking transition numerically, the authors track the divergence of the Fourier transform of the correlation function at zero frequency, \hat C(0). In equilibrium (passive) settings, this divergence follows a known power law, facilitating accurate extrapolation of transition energies.

In the persistent active setting studied here, the relevant critical power-law behavior is not established, preventing precise measurement of the asymptotic energy via \hat C(0) fits. Identifying this exponent would improve accuracy and generality of transition energy determinations for active walkers.