We investigate the short-time regime of the KPZ equation in
1+11+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})P(H,t)∼exp(−Φ(H)/t)
for small time. We obtain the rate function
\Phi(H)Φ(H)
analytically for the above cases. It has a Gaussian form in the center
with asymmetric tails, |H|^{5/2}|H|5/2
on the negative side, and H^{3/2}H3/2
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.