The scaling limit of the longest increasing subsequence

Abstract: We provide a framework for proving convergence to the directed landscape, the central object in the Kardar-Parisi-Zhang universality class. For last passage models, we show that compact convergence to the Airy line ensemble implies convergence to the Airy sheet. In i.i.d. environments, we show that Airy sheet convergence implies convergence of distances and geodesics to their counterparts in the directed landscape. Our results imply convergence of classical last passage models and interacting particle systems. Our framework is built on the notion of a directed metric, a generalization of metrics which behaves better under limits. As a consequence of our results, we present a solution to an old problem: the scaled longest increasing subsequence in a uniform permutation converges to the directed geodesic.