Replica approach to the KPZ equation with the half Brownian motion initial condition

Abstract

We consider the one-dimensional Kardar–Parisi–Zhang (KPZ) equation with the half Brownian motion initial condition, studied previously through the weakly asymmetric simple exclusion process. We employ the replica Bethe ansatz and show that the generating function of the exponential moments of the height is expressed as a Fredholm determinant. From this, the height distribution and its asymptotics are studied. Furthermore, using the replica method we also discuss the multi-point height distribution. We find that some good properties of the deformed Airy functions play an important role in the analysis.