The directed landscape from Brownian motion

TL;DR

This paper constructs a bijection from Brownian motions to the directed landscape, providing explicit mappings, convergence results, and applications to reconstructing the landscape from the Airy line ensemble.

Contribution

It introduces a natural scaling limit of RSK correspondence from Brownian motions to the directed landscape, with explicit inverse maps and new semi-discrete RSK versions.

Findings

Constructed an almost sure bijection from Brownian motions to the directed landscape.

Proved convergence of Brownian last-passage percolation to the directed landscape.

Reconstructed the landscape from the parabolic Airy line ensemble, confirming a conjecture.

Abstract

We construct an almost sure bijection that recovers the directed landscape on the half-plane from a sequence of independent Brownian motions. This map is the natural scaling limit of the Robinson--Schensted--Knuth (RSK) correspondence. The Brownian motions arise as the marginals of the multi-path stationary horizon associated with the directed landscape. The inverse map is fully explicit and yields a natural coupling in which Brownian last-passage percolation converges in probability to the directed landscape. As an application, we prove that the directed landscape restricted to a strip can be reconstructed from the parabolic Airy line ensemble, resolving a conjecture of the first author and Zhang. Along the way we develop two new versions of RSK in the semi-discrete setting, introduce a general theory of sorting via Pitman operators that generates a faithful action of the biHecke monoid, and establish key identities for the multi-path stationary horizon for both the directed landscape and Brownian last-passage percolation.