Non-Markovian derivation of the double distribution and universal-scaling profile

Derive the double distribution within a non-Markovian excursion-set framework, such as the path-integral formulation of Maggiore and Riotto or the Fokker–Planck framework of López-Díaz et al., and establish whether this derivation explains how the universal-scaling outer density profile can fit simulations despite its disagreement with the classical excursion-set prediction.

Background

The paper attributes the discrepancy between the classical excursion-set prediction and N-body outer density profiles to the Markovian assumption induced by the sharp k-space window function. In particular, the conditional-probability substitution used in deriving the double distribution may fail for correlated, non-Markovian density trajectories.

The authors identify existing non-Markovian formalisms, including the path-integral approach of Maggiore and Riotto and the Fokker–Planck approach of López-Díaz et al., as possible foundations for a more rigorous re-derivation. The unresolved calculation is intended to explain why the universal-scaling profile, obtained from the classical framework with simplifying assumptions, nevertheless provides a good fit to simulations.