Short-time height distribution in 1d KPZ equation: starting from a parabola

We study the probability distribution $\mathcal{P}(H,t,L)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension when starting from a parabolic interface, $h(x,t=0)=x^2\mathrm{/}L$. The limits of $L\to \mathrm{\infty }$ and $L\to 0$ have been recently solved exactly for any $t>0$. Here we address the early-time behavior of $\mathcal{P}(H,t,L)$ for general $L$. We employ the weak-noise theory - a variant of WKB approximation – which yields the optimal history of the interface, conditioned on reaching the given height $H$ at the origin at time $t$. We find that at small $H$ $\mathcal{P}(H,t,L)$ is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as $-\ln \mathcal{P}=f_{+}\mathrm{\mid }H{\mathrm{\mid }}^{5\mathrm{/}2}\mathrm{/}{t}^{1\mathrm{/}2}$ and $f_{-}\mathrm{\mid }H{\mathrm{\mid }}^{3\mathrm{/}2}\mathrm{/}{t}^{1\mathrm{/}2}$. The factor $f_{+}(L,t)$ monotonically increases as a function of $L$, interpolating between time-independent values at $L=0$ and $L=\mathrm{\infty }$ that were previously known. The factor $f_{-}$ is independent of $L$ and $t$, signalling universality of this tail for a whole class of deterministic initial conditions.