The One-dimensional KPZ Equation and the Airy Process

TL;DR

This paper extends previous work on the 1D KPZ equation to joint height statistics at multiple points, showing convergence to the Airy process distribution via Fredholm determinants.

Contribution

It introduces a method to compute the joint height distribution for the KPZ equation at multiple points and links it to the Airy process through Fredholm determinants.

Findings

Convergence of the n-point generating function to the Airy process distribution.

Explicit expression of the joint height statistics using Fredholm determinants.

Validation of the factorization assumption for the joint distribution.

Abstract

Our previous work on the one-dimensional KPZ equation with sharp wedge initial data is extended to the case of the joint height statistics at n spatial points for some common fixed time. Assuming a particular factorization, we compute an n-point generating function and write it in terms of a Fredholm determinant. For long times the generating function converges to a limit, which is established to be equivalent to the standard expression of the n-point distribution of the Airy process.