An exact solution for the KPZ equation with flat initial conditions

TL;DR

This paper presents the first exact solution for the height distribution of the 1D KPZ equation with flat initial conditions at any time, using advanced mathematical techniques and showing convergence to the GOE Tracy-Widom distribution.

Contribution

It provides an exact formula for the KPZ height distribution with flat initial conditions at all times, utilizing the Bethe Ansatz and Fredholm Pfaffian methods.

Findings

Distribution converges to GOE Tracy-Widom at large times

Exact formula valid for all times

Uses Bethe Ansatz and Fredholm Pfaffian techniques

Abstract

We provide the first exact calculation of the height distribution at arbitrary time $t$ of the continuum KPZ growth equation in one dimension with flat initial conditions. We use the mapping onto a directed polymer (DP) with one end fixed, one free, and the Bethe Ansatz for the replicated attractive boson model. We obtain the generating function of the moments of the DP partition sum as a Fredholm Pfaffian. Our formula, valid for all times, exhibits convergence of the free energy (i.e. KPZ height) distribution to the GOE Tracy Widom distribution at large time.