When the geodesic becomes rigid in the directed landscape

TL;DR

This paper investigates the behavior of geodesics in the directed landscape when the landscape's value is large, revealing rigidity and Gaussian fluctuation properties of geodesic locations.

Contribution

It establishes the conditions under which geodesics become rigid and characterizes their fluctuations, providing new insights into the geometry of the directed landscape.

Findings

Geodesics become rigid when the landscape value is large.

Geodesic fluctuations around the expectation are of order L^{-1/4}.

Midpoint fluctuations converge to independent Gaussian distributions.

Abstract

When the value $L$ of the directed landscape at a point $(\mathbf{p};\mathbf{q})$ is sufficiently large, the geodesic from $\mathbf{p}$ to $\mathbf{q}$ 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.