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We consider the fluctuations in the stochastic growth of a one-dimensionalinterface 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 initialconditions at $t=0$. In the limit of large times this JPDF is expected tobecome a universal function of the time ratio $t_2/t_1$, and of the (properlyscaled) heights $h(x,t_1)$ and $h(x,t_2)$. Using the replica Bethe ansatzmethod for the KPZ equation, in [J. Stat. Mech. (2017) 053212] we obtained aformula for the JPDF in the (partial) tail regime where $h(x,t_1)$ is large andpositive, subsequently found in excellent agreement with experimental andnumerical data [Phys. Rev. Lett. 118, 125701 (2017)]. Here we show that ourresults are in perfect agreement with Johansson's recent rigorous expression ofthe full JPDF [arXiv:1802.00729 ], thereby confirming the validity of ourmethods.

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