When the geodesic becomes rigid in the directed landscape

Abstract

When the value L of the directed landscape at a point ( p ; 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.