Title: Exact short-time height distribution in 1D KPZ equation with Brownian initial condition,
Abstract: The early time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$
dimension, starting from a Brownian initial condition with a drift $w$, is
studied using the exact Fredholm determinant representation. For large drift we
recover the exact results for the droplet initial condition, whereas a
vanishingly small drift describes the stationary KPZ case, recently studied by
weak noise theory (WNT). We show that for short time $t$, the probability
distribution $P(H,t)$ of the height $H$ at a given point takes the large
deviation form $P(H,t) \sim \exp{\left(-\Phi(H)/\sqrt{t} \right)}$. We obtain
the exact expressions for the rate function $\Phi(H)$ for $H<H_{c2}$. Our exact
expression for $H_{c2}$ numerically coincides with the value at which WNT was
found to exhibit a spontaneous reflection symmetry breaking. We propose two
continuations for $H>H_{c2}$, which apparently correspond to the symmetric and
asymmetric WNT solutions. The rate function $\Phi(H)$ is Gaussian in the
center, while it has asymmetric tails, $|H|^{5/2}$ on the negative $H$ side and
$H^{3/2}$ on the positive $H$ side.