A non-trivial dynamics on the directed landscape

Abstract

We prove the existence of a non-trivial continuous stationary reversible dynamics on the directed landscape as a subsequential scaling limit of Brownian last passage percolation under the Ornstein--Uhlenbeck dynamics. Strong passage-time stability estimates from the companion paper Bhatia '26, together with static endpoint regularity, give tightness at the critical dynamical scale $n^{-1/3}$. To establish non-triviality of the subsequential limiting dynamics, we show that, in dynamical BLPP, the fractional overlap between geodesics at times zero and $a n^{-1/3}$ tends to one as first $n\to\infty$ and then $a\downarrow0$. The proof refines the excursion argument of Ganguly-Hammond '24, using strong passage-time stability and the fact that a fixed dynamical time is increasingly subcritical at finer spatial scales.