The one-dimensional KPZ equation: an exact solution and its universality

TL;DR

This paper presents the first exact solution to the one-dimensional KPZ equation, revealing its connection to Tracy-Widom distribution and confirming its role in describing interface motion under weak driving force.

Contribution

It provides an exact determinantal formula for the height distribution in the 1D KPZ equation, establishing its universality and link to random matrix theory.

Findings

Exact solution for the 1D KPZ equation with curved initial condition

Large time height distribution follows Tracy-Widom distribution

Confirms KPZ equation's role in interface motion under weak driving force

Abstract

We report on the first exact solution of the KPZ equation in one dimension, with an initial condition which physically corresponds to the motion of a macroscopically curved height profile. The solution provides a determinantal formula for the probability distribution function of the height $h(x,t)$ for all $t>0$. In particular, we show that for large $t$, on the scale $t^{1/3}$, the statistics is given by the Tracy-Widom distribution, known already from the theory of GUE random matrices. Our solution confirms that the KPZ equation describes the interface motion in the regime of weak driving force. Within this regime the KPZ equation details how the long time asymptotics is approached.