We construct an almost sure bijection that recovers the directed landscape on the half-plane from a sequence of independent Brownian motions. This is the natural scaling limit of the Robinson–Schensted–Knuth (RSK) correspondence.

The map sends the directed landscape to the multi-path stationary horizon, whose marginals are independent Brownian motions. The inverse is fully explicit and realized through Busemann shears on Brownian last-passage percolation. 

Under this coupling, the drifted environments converge in probability to the directed landscape.

As an application, we resolve a conjecture of Dauvergne and Zhang by showing that the directed landscape on any time strip can be reconstructed from the parabolic Airy line ensemble.

Joint work with Duncan Dauvergne.