The KPZ equation and the directed landscape

TL;DR

This paper establishes the first convergence of the KPZ equation's narrow wedge solutions to the Airy sheet and directed landscape, advancing understanding of positive temperature models in stochastic growth.

Contribution

It proves convergence to the Airy sheet and directed landscape for the KPZ equation, and provides an independent proof of convergence to the KPZ fixed point for general initial conditions.

Findings

Convergence of narrow wedge solutions to the Airy sheet.

Convergence to the directed landscape in the locally uniform topology.

Joint convergence to the KPZ fixed point for multiple initial conditions.

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.