Characterization of the directed landscape from the KPZ fixed point

Abstract

We show that the directed landscape is the unique coupling of the KPZ fixed point from all initial conditions satisfying three natural properties: independent increments, monotonicity, and shift commutativity. Equivalently, we show that the directed landscape is the unique directed metric on $\mathbb R^2$ with independent increments and KPZ fixed point marginals. This unifies the two central objects in the KPZ universality class. Our main theorem also provides a general framework for proving convergence to the directed landscape given convergence to the KPZ fixed point. We apply this framework to prove landscape convergence in a range of models: exotic couplings of ASEP and TASEP, the random walk and Brownian web distances, and a class of non-integrable asymmetric exclusion processes with the basic coupling that perturb off of TASEP (this final class requires random initial data). All of our convergence theorems are new except for colored TASEP, where we provide a short alternative proof.