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

Abstract

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 $A⩾-1/2$, 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 $A=-\frac{1}{2}$.

43 pages

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• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• The lower tail of the half-space KPZ equation
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Lyapunov exponents of the half-line SHE
• Boundary current fluctuations for the half space ASEP and six vertex model
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case