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

TL;DR

This paper analyzes the joint probability distribution of height fluctuations in 1D KPZ growth at two different times, confirming previous results with a rigorous expression and exploring the tail behavior using replica methods.

Contribution

The authors validate their replica Bethe ansatz results against Johansson's rigorous formula for the two-time height distribution in KPZ growth.

Findings

Agreement with Johansson's exact formula

Confirmation of replica method validity

Tail regime behavior characterized

Abstract

We consider the fluctuations in the stochastic growth of a one-dimensional interface of height $h(x,t)$ described by the Kardar-Parisi-Zhang (KPZ) universality class. We study the joint probability distribution function (JPDF) of the interface heights at two times $t_1$ and $t_2>t_1$, with droplet initial conditions at $t=0$. In the limit of large times this JPDF is expected to become a universal function of the time ratio $t_2/t_1$, and of the (properly scaled) heights $h(x,t_1)$ and $h(x,t_2)$. Using the replica Bethe ansatz method for the KPZ equation, in [J. Stat. Mech. (2017) 053212] we obtained a formula for the JPDF in the (partial) tail regime where $h(x,t_1)$ is large and positive, subsequently found in excellent agreement with experimental and numerical data [Phys. Rev. Lett. 118, 125701 (2017)]. Here we show that our results are in perfect agreement with Johansson's recent rigorous expression of the full JPDF [arXiv:1802.00729 ], thereby confirming the validity of our methods.