The starting point of a directed geodesic is special

Abstract

For any $t\in [0,1/2)$, consider the process ${\eta }_t:[0,1/2]\mapsto \mathbb{R}$ defined by ${\eta }_t(s)=\Pi (t+s)-\Pi (t)$, where $\Pi$ is the directed geodesic from $(0,0)$ to $(0,1)$ in the directed landscape. Let $0\le t<u<1/2$. We show that the laws of ${\eta }_t$ and ${\eta }_u$ are mutually absolutely continuous if and only if $t>0$. This proves Conjecture 14.5 in Dauvergne, Ortmann, and Virág, 2022 in the affirmative.