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 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 nonanalytic. The analytic branch is nonphysical 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 $S_{\text{f}}\left(H\right)=2^{-3/2}{S}_{\text{st}}\left(2H\right)$.