Characterization of the extensive-duration big-jump regime

Characterize the distribution of the logarithmic observable A=-\sum_{i=1}^{N}\ln x_i for the doubling, tent, and logistic chaotic maps in the regime A\sim N^2, where the big-jump excursion occupies an O(1) fraction of the trajectory and is no longer a localized event.

Background

The paper derives the stretched-exponential right tail of the observable when a single localized excursion near the fixed point x=0 produces the excess A_1. This analysis assumes that the duration M of the excursion satisfies M\ll N, equivalently A_1,A\ll N2, so that the condensed contribution occupies only a small portion of the trajectory.

At fluctuations of order A\sim N2, the excursion duration becomes comparable to the total trajectory length, invalidating the localized-event approximation and requiring a different description. The authors explicitly defer this regime, which is analogous to regimes studied in related stochastic systems.

References

In this subsection, we tacitly assumed that M \ll N, or equivalently that A_1, A \ll N2, so that the duration of the “big jump” is much shorter than the duration of the entire dynamics. In the regime of very large fluctuations, A \sim N2, the big jump occupies an O(1) fraction of the dynamics and is no longer a localized event. Analogous regimes have been studied in other systems , but we leave this regime for future work.

— Dynamical phase transition for a logarithm observable in chaotic systems  (2610.21288 - Defaveri et al., 15 Oct 2026) in Section 3.1, “Stretched exponential tail — leading order”