Crossover between various initial conditions in KPZ growth: flat to stationary

Abstract

We conjecture the universal probability distribution at large time for the one-point height in the 1D Kardar-Parisi-Zhang (KPZ) stochastic growth universality class, with initial conditions interpolating from any one of the three main classes (droplet, flat, stationary) on the left, to another on the right, allowing for drifts and also for a step near the origin. The result is obtained from a replica Bethe ansatz calculation starting from the KPZ continuum equation, together with a "decoupling assumption" in the large time limit. Some cases are checked to be equivalent to previously known results from other models in the same class, which provides a test of the method, others appear to be new. In particular we obtain the crossover distribution between flat and stationary initial conditions (crossover from Airy$_1$ to Airy$_{\rm stat}$) in a simple compact form.