Characterization of the directed landscape from the KPZ fixed point

Duncan Dauvergne, Lingfu Zhang

Published 2024-12-17Version 1

We show that the directed landscape is the unique coupling of the KPZ fixed point from all initial conditions at all times 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 gives a framework for proving convergence to the directed landscape given convergence to the KPZ fixed point. We apply this framework to prove convergence to the directed landscape for a range of models, some without exact solvability: asymmetric exclusion processes with potentially non-nearest neighbour interactions, exotic couplings of ASEP, the random walk and Brownian web distance, and directed polymer models. All of our convergence theorems are new except for colored ASEP and the KPZ equation, where we provide alternative proofs.