Scaling in Stochastic Ghosts

The paper develops a WKB/Hamiltonian picture for one‑dimensional birth–death systems near a saddle‑node and shows (via phase‑space analysis and numerics) that, for small φ = ε − εc, statistically significant trajectories have negative initial momentum p(t0) < 0 and O(1) flight times, while p(t0) ≥ 0 trajectories scale like φ−1/2 but carry negligible weight; this underpins a universal shape of the scaling function G(·) across models. These claims are clearly stated and illustrated (Hamiltonian and path integral setup; selection of p < 0; φ−1/2 for p ≥ 0; universality claim) and supported by figures and appendix analysis, though not proved rigorously . The candidate solution gives a compatible but more formal normal‑form argument: it identifies the activation branch p = ln(w−/w+) < 0 on H = 0, shows p = 0 is invariant, performs an inner rescaling y = √φ Y, p = √φ P to get O(1) times for p < 0 and φ−1/2 for p ≥ 0, and argues a dominance of the p < 0 family. Minor technical slips (e.g., the scaling of H and of Ṗ in the inner layer) do not alter the main conclusions; the paper’s narrative does not hinge on these details. Overall, the two are in agreement on the mechanism and predicted scaling/selection; the model adds analytic structure, while the paper provides the empirical/numerical backbone.