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The KPZ equation and the directed landscape

Xuan Wu

TL;DR

The paper proves that narrow wedge solutions to the KPZ equation converge to the Airy sheet and the directed landscape in the locally uniform topology, marking the first such positive-temperature convergence to these universal KPZ objects.A central component is the 1:2:3 scaling of KPZ objects and the demonstration that scaled KPZ sheets converge to Airy sheets, which then assemble into the directed landscape via a variational max-convolution structure.An independent proof is provided for convergence of KPZ to the KPZ fixed point for general initial data, and the results extend to joint convergence for multiple initial conditions, enabling process-level understanding of large-time KPZ dynamics.The approach weaves together semi-discrete polymers, the geometric RSK correspondence, and the KPZ line ensemble, exploiting a key invariance and concentration phenomena to bridge polymer energies, line ensembles, and the Airy/landscape limit.

Abstract

This paper proves the convergence of the narrow wedge solutions of the KPZ equation to the Airy sheet and the directed landscape in the locally uniform topology. This is the first convergence result to the Airy sheet and the directed landscape established for a positive temperature model. We also give an independent proof for the convergence of the KPZ equation to the KPZ fixed point for general initial conditions in the locally uniform topology. Together with the directed landscape convergence, we show the joint convergence to the KPZ fixed point for multiple initial conditions.

The KPZ equation and the directed landscape

TL;DR

The paper proves that narrow wedge solutions to the KPZ equation converge to the Airy sheet and the directed landscape in the locally uniform topology, marking the first such positive-temperature convergence to these universal KPZ objects.A central component is the 1:2:3 scaling of KPZ objects and the demonstration that scaled KPZ sheets converge to Airy sheets, which then assemble into the directed landscape via a variational max-convolution structure.An independent proof is provided for convergence of KPZ to the KPZ fixed point for general initial data, and the results extend to joint convergence for multiple initial conditions, enabling process-level understanding of large-time KPZ dynamics.The approach weaves together semi-discrete polymers, the geometric RSK correspondence, and the KPZ line ensemble, exploiting a key invariance and concentration phenomena to bridge polymer energies, line ensembles, and the Airy/landscape limit.

Abstract

This paper proves the convergence of the narrow wedge solutions of the KPZ equation to the Airy sheet and the directed landscape in the locally uniform topology. This is the first convergence result to the Airy sheet and the directed landscape established for a positive temperature model. We also give an independent proof for the convergence of the KPZ equation to the KPZ fixed point for general initial conditions in the locally uniform topology. Together with the directed landscape convergence, we show the joint convergence to the KPZ fixed point for multiple initial conditions.
Paper Structure (17 sections, 49 theorems, 297 equations, 1 figure)

This paper contains 17 sections, 49 theorems, 297 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kardar-Parisi-Zhang equation
  3. The narrow wedge solution
  4. The Airy line ensemble, Airy sheets and the directed landscape
  5. Main results
  6. O'Connell Yor polymers and the KPZ line ensemble
  7. Ingredients and ideas
  8. Outline
  9. Notation
  10. Acknowledgement
  11. Semi-Discrete Polymers
  12. Geometric RSK correspondence
  13. O’Connell-Yor polymer model
  14. Proof of Theorem \ref{['thm:KPZtoAiry_sheet']}
  15. Proof of Theorem \ref{['thm:KPZtoLandscape']}
  16. ...and 2 more sections

Key Result

Proposition 1.1

There exists a coupling of with the following properties.

Figures (1)

  • Figure 1: The right and the bottom arrows are two major inputs. The top arrow is one of our main result, Theorem \ref{['thm:KPZtoAiry_sheet']}. We prove it through establishing the left arrow, a relation between the KPZ sheet and the KPZ line ensemble.

Theorems & Definitions (97)

  • Proposition 1.1: AJRS
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Proposition 1.10: CH16Nic21
  • ...and 87 more