  We study the long-time regime of the Kardar-Parisi-Zhang (KPZ) equation in
$1+1$ dimensions for the Brownian and droplet initial conditions and present a
simple derivation of the tail of the large deviations of the height on the
negative side $\lambda H<0$. We show that for both initial conditions, the
cumulative distribution functions take a large deviations form, with a tail for
$- \tilde s \gg 1$ given by $-\log
\mathbb{P}\left(\frac{H}{t}<\tilde{s}\right)=t^2 \frac{4 }{15 \pi}
(-\tilde{s})^{5/2} $. This exact expression was already observed at small time
for both initial conditions suggesting that these large deviations remain valid
at all times. We present two methods to derive the result (i) long time
estimate using a Fredholm determinant formula and (ii) the evaluation of the
cumulants of a determinantal point process where the successive cumulants
appear to give the successive orders of the large deviation rate function in
the large $\tilde s$ expansion. An interpretation in terms of large deviations
for trapped fermions at low temperature is also given. In addition, we perform
a similar calculation for the KPZ equation in a half-space with a droplet
initial condition, and show that the same tail as above arises, with the
prefactor $\frac{4}{15\pi}$ replaced by $\frac{2}{15\pi}$. Finally, the
arguments can be extended to show that this tail holds for all times. This is
consistent with the fact that the same tail was obtained previously in the
short time limit for the full-space problem.
