We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic
growth of an interface of height h(x,t)h(x,t)
on the positive half line with boundary condition
\partial_x h(x,t)|_{x=0}=A∂xh(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=0x=0
either repulsive A>0A>0,
or attractive A<0A<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 -1/2A≥−1/2,
and the Brownian initial condition with a drift for
A=+\inftyA=+∞
(infinite hard wall). We study the height at
x=0x=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}A>−12
the large time PDF is the GSE Tracy-Widom distribution. For
A= \frac{1}{2}A=12,
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 0A+12=ϵt−1/3→0
with fixed \epsilon = \mathcal{O}(1)ϵ=𝒪(1),
we obtain a transition kernel continuously depending on
\epsilonϵ.
Our work extends the results obtained previously for
A=+\inftyA=+∞,
A=0A=0
and A=- \frac{1}{2}A=−12.
Sufficient Condition for Wavelet Frame on Positive Half-Line