When the geodesic becomes rigid in the directed landscape

Abstract

When the value L of the directed landscape at a point $(\mathbf{p};\mathbf{q})$ is sufficiently large, the geodesic from p to q is rigid and its location fluctuates of order $L^{-1∕4}$ around its expectation. We further show that at a midpoint of the geodesic, the location of the geodesic and the value of the directed landscape after appropriate scaling converge to two independent Gaussians.