Acta Mathematica

Volume 229 (2022)

Number 2

The directed landscape

Pages: 201 – 285



Duncan Dauvergne (Department of Mathematics, University of Toronto, Ontario, Canada)

Janosch Ortmann (Département de management et technologie, École des sciences de gestion, Université du Québec à Montréal, Canada)

Bálint Virág (Departments of Mathematics and Statistics, University of Toronto, Ontario, Canada)


The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random functions with Hölder-$\frac{2}{3}^-$ continuous paths. This work completes the construction of the central object in the Kardar–Parisi–Zhang universality class, the directed landscape.

D.D. was supported by an NSERC CGS D scholarship.

B.V. was supported by the Canada Research Chair program, the NSERC Discovery Accelerator grant, the MTA Momentum Random Spectra research group, and the ERC consolidator grant 648017 (Abert).

Received 26 June 2019

Received revised 7 May 2021

Accepted 7 June 2021

Published 21 February 2023