SciPost Phys. 8, 035 (2020) ·
published 4 March 2020
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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 $A=- \frac{1}{2}$.
SciPost Phys. 5, 032 (2018) ·
published 12 October 2018
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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.
