Limiting one-point fluctuations of the geodesic in the directed landscape near the endpoints when the geodesic length goes to infinity

We consider the limiting fluctuations of the geodesic in the directed landscape, conditioning on its length going to infinity. It was shown in \cite{Liu22b,Ganguly-Hegde-Zhang23} that when the directed landscape $\mathcal{L}(0,0;0,1) = L$ becomes large, the geodesic from $(0,0)$ to $(0,1)$ lies in a strip of size $O(L^{-1/4})$ and behaves like a Brownian bridge if we zoom in the strip by a factor of $L^{1/4}$. Moreover, the length along the geodesic with respect to the directed landscape fluctuates of order $O(L^{1/4})$ and its limiting one-point distribution is Gaussian \cite{Liu22b}. In this paper, we further zoom in a smaller neighborhood of the endpoints when $\mathcal{L}(0,0;0,1) = L$ or $\mathcal{L}(0,0;0,1) \ge L$, and show that there is a critical scaling window $L^{-3/2}:L^{-1}:L^{-1/2}$ for the time, geodesic location, and geodesic length, respectively. Within this scaling window, we find a nontrivial limit of the one-point joint distribution of the geodesic location and length as $L\to\infty$. This limiting distribution, if we tune the time parameter to infinity, converges to the joint distribution of two independent Gaussian random variables, which is consistent with the results in \cite{Liu22b}. We also find a surprising connection between this limiting distribution and the one-point distribution of the upper tail field of the KPZ fixed point recently obtained in \cite{Liu-Zhang25}.

💡 Research Summary

The paper investigates the fine‑scale fluctuations of the geodesic in the directed landscape under the “upper‑tail” conditioning that the passage time $\mathcal L(0,0;0,1)$, denoted $L$, tends to infinity. Earlier works (Liu 2022b, Ganguly‑Hegde‑Zhang 2023) showed that for a fixed macroscopic time $t\in(0,1)$ the geodesic $\pi(t)$ stays inside a strip of width $O(L^{-1/4})$ and, after scaling space by $L^{1/4}$, converges to a standard Brownian bridge. Moreover, the length of the geodesic segment $L(t)$ fluctuates on the $O(L^{1/4})$ scale and its one‑point distribution is Gaussian.

The present work zooms in much closer to the endpoints. The authors identify a critical scaling window \