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 [Liu22c, GHZ23] that when the directed landscape 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 L1∕4. Moreover, the length along the geodesic with respect to the directed landscape fluctuates of order O(L1∕4) and its limiting one-point distribution is Gaussian [Liu22c]. In this paper, we further zoom in a smaller neighborhood of the endpoints when L(0,0;0,1)=L or L(0,0;0,1)≥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→∞. 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 [Liu22c]. 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 [LZ25].