The scaling limit of the longest increasing subsequence

TL;DR

This paper develops a framework for proving convergence of last passage models and related systems to the directed landscape, establishing the limiting behavior of the longest increasing subsequence in uniform permutations.

Contribution

It introduces a new framework based on directed metrics for proving convergence to the directed landscape, solving the longstanding problem of the LIS in uniform permutations.

Findings

Convergence of classical last passage models to the directed landscape.

Convergence of geodesics and distances in i.i.d. environments.

Longest increasing subsequence in uniform permutations converges to the directed geodesic.

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.