The directed landscape

2022/01/01 by Duncan Dauvergne, Janosch Ortmann, Bálint Virág · 60 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Brownian motion #Geodesic #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Random Matrices and Applications #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Universality (dynamical systems)

paper · pdf · doi:10.4310/acta.2022.v229.n2.a1

published in Acta Mathematica 229(2), 201-285 (Mittag-Leffler Institute)

openalex publication_date 2022/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 Holder-2/3- continuous paths. This work completes the construction of the central object in the Kardar-Parisi-Zhang universality class, the directed landscape.

Citations

• The KPZ fixed point

Cited by

• Shift‐invariance for vertex models and polymers
• Critical fluctuations of last passage percolation with thick boundaries
• Last passage percolation in hierarchical environments
• Decoupling and decay of two-point functions in a two-species (T)ASEP
• Two-time spatial decorrelation for the flat KPZ fixed point
• Invariant measures and shocks in the KPZ fixed point
• Structural properties of the Airy wanderer line ensembles
• On the convoy of the ASEP speed process
• van den Berg-Kesten--type correlation inequalities for disjoint polymers in the KPZ universality class
• Law of Large Numbers for continuous N-particle ensembles at fixed temperature
• Periodic TASEP with general initial conditions
• Convergence of the Environment Seen from Geodesics in Exponential Last-Passage Percolation
• Near-the-Axis Universality in Last Passage Percolation
• Tightness of (H, H^RW)-Gibbsian line ensembles
• KPZ on torus: Gaussian fluctuations
• Quantitative Brownian regularity of the KPZ fixed point with arbitrary initial data
• Radon-Nikodym derivative of inhomogeneous Brownian last passage percolation
• Bilinear Differential-Difference Equations and One-Point Distributions of Some KPZ-Class Models
• Multipoint distributions of the KPZ fixed point with compactly supported initial conditions
• Limiting one-point fluctuations of the geodesic in the directed landscape near the endpoints when the geodesic length goes to infinity
• The discrete periodic Pitman transform: invariances, braid relations, and Burke properties
• On the multipoint distribution formulas of the parabolic Airy process
• The heat and the landscape I
• Universality of deterministic KPZ
• Multi-point distribution of discrete time periodic TASEP
• Dynamic scaling of growing interfaces
• Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison
• Brownian regularity for the KPZ line ensemble
• When the geodesic becomes rigid in the directed landscape
• Convergence from the Log-Gamma Polymer to the Directed Landscape
• Disjoint optimizers and the directed landscape
• Tightness of Bernoulli Gibbsian line ensembles
• Revisiting the temporal law in KPZ random growth
• Edge Universality for Nonintersecting Brownian Bridges
• GUE Fluctuations Near the Axis in One-Sided Ballistic Deposition
• Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials
• Near-existence of bigeodesics in dynamical exponential last passage percolation
• Anomalous Geodesics in the Inhomogeneous Corner Growth Model. [europepmc]

Related

• Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape
• Disjoint optimizers and the directed landscape
• The scaling limit of the longest increasing subsequence
• The directed landscape from Brownian motion
• Near-the-Axis Universality in Last Passage Percolation