Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica

Abstract

We consider the fluctuations in the stochastic growth of a one-dimensional interface of height described by the Kardar–Parisi–Zhang (KPZ) universality class. We study the joint probability distribution function (JPDF) of the interface heights at two-times t1 and , with droplet initial conditions at t = 0. In the large-time limit, this JPDF is expected to become a universal function of the time ratio , and of the (properly scaled) heights h(x,t1) and h(x,t2). Using the replica Bethe ansatz method for the KPZ equation, in De Nardis and Le Doussal (2017 J. Stat. Mech. 053212) we obtained a formula for the JPDF in the (partial) tail regime where h(x,t1) is large and positive, subsequently found to be in excellent agreement with the experimental and numerical data (De Nardis et al 2017 Phys. Rev. Lett. 118 125701). Here we show that our results are in perfect agreement with Johansson’s recent rigorous expression of the full JPDF (Johansson 2018 in preparation (arXiv:1802.00729)), thereby confirming the validity of our methods.