Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface

We determine the exact short-time distribution $-\ln \mathcal{P}_{\text{f}}\left(H,t\right)= S_{\text{f}} \left(H\right)/\sqrt{t}$ of the one-point height $H=h(x=0,t)$ of an evolving 1+1 Kardar-Parisi-Zhang (KPZ) interface for flat initial condition. This is achieved by combining (i) the optimal fluctuation method, (ii) a time-reversal symmetry of the KPZ equation in 1+1 dimension, and (iii) the recently determined exact short-time height distribution $-\ln \mathcal{P}_{\text{st}}\left(H,t\right)= S_{\text{st}} \left(H\right)/\sqrt{t}$ for \emph{stationary} initial condition. In studying the large-deviation function $S_{\text{st}} \left(H\right)$ of the latter, one encounters two branches: an analytic and a non-analytic. The analytic branch is non-physical beyond a critical value of $H$ where a second-order dynamical phase transition occurs. Here we show that, remarkably, it is the analytic branch of $S_{\text{st}} \left(H\right)$ which determines the large-deviation function $S_{\text{f}} \left(H\right)$ of the flat interface via a simple mapping $S_{\text{f}}\left(H\right)=2^{-3/2}S_{\text{st}}\left(2H\right)$.