Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

Understand

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).