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

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 ∂ 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 x = 0 either repulsive A>0 A > 0 , or attractive A - \frac{1}{2} A > − 1 2 the large time PDF is the GSE Tracy-Widom distribution. For A= \frac{1}{2} A = 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 A + 1 2 = ϵ t − 1 / 3 → 0 with fixed \epsilon = \mathcal{O}(1) ϵ = ( 1 ) , we obtain a transition kernel continuously depending on \epsilon ϵ . Our work extends the results obtained previously for A=+\infty A = + ∞ , A=0 A = 0 and A=- \frac{1}{2} A = − 1 2 .