| Summary: | We investigate the short-time regime of the KPZ equation in $1+1$ dimensions
and develop a unifying method to obtain the height distribution in this regime,
valid whenever an exact solution exists in the form of a Fredholm Pfaffian or
determinant. These include the droplet and stationary initial conditions in
full space, previously obtained by a different method. The novel results
concern the droplet initial condition in a half space for several Neumann
boundary conditions: hard wall, symmetric, and critical. In all cases, the
height probability distribution takes the large deviation form $P(H,t) \sim
\exp( - \Phi(H)/\sqrt{t})$ for small time. We obtain the rate function
$\Phi(H)$ analytically for the above cases. It has a Gaussian form in the
center with asymmetric tails, $|H|^{5/2}$ on the negative side, and $H^{3/2}$
on the positive side. The amplitude of the left tail for the half-space is
found to be half the one of the full space. As in the full space case, we find
that these left tails remain valid at all times. In addition, we present here
(i) a new Fredholm Pfaffian formula for the solution of the hard wall boundary
condition and (ii) two Fredholm determinant representations for the solutions
of the hard wall and the symmetric boundary respectively.
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