| Summary: | We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth
of an interface of height $h(x,t)$ on the positive half line with boundary
condition $\partial_x h(x,t)|_{x=0}=A$. It is equivalent to a continuum
directed polymer (DP) in a random potential in half-space with a wall at $x=0$
either repulsive $A>0$, or attractive $A<0$. We provide an exact solution,
using replica Bethe ansatz methods, to two problems which were recently proved
to be equivalent [Parekh, arXiv:1901.09449]: the droplet initial condition for
arbitrary $A \geqslant -1/2$, and the Brownian initial condition with a drift
for $A=+\infty$ (infinite hard wall). We study the height at $x=0$ and obtain
(i) at all time the Laplace transform of the distribution of its exponential
(ii) at infinite time, its exact probability distribution function (PDF). These
are expressed in two equivalent forms, either as a Fredholm Pfaffian with a
matrix valued kernel, or as a Fredholm determinant with a scalar kernel. For
droplet initial conditions and $A> - \frac{1}{2}$ the large time PDF is the GSE
Tracy-Widom distribution. For $A= \frac{1}{2}$, the critical point at which the
DP binds to the wall, we obtain the GOE Tracy-Widom distribution. In the
critical region, $A+\frac{1}{2} = \epsilon t^{-1/3} \to 0$ with fixed $\epsilon
= \mathcal{O}(1)$, we obtain a transition kernel continuously depending on
$\epsilon$. Our work extends the results obtained previously for $A=+\infty$,
$A=0$ and $A=- \frac{1}{2}$.
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