An Invariance Principle of 1D KPZ with Robin Boundary Conditions

We consider a discrete one-dimensional random interface on the half-space whose height at any positive point is composed of a function of the heights at its two closest neighbours and an independent random noise background. In [AC24], Adhikari and Chatterjee proved for the full-space model that the height function of such a Markov process converges to the Cole-Hopf solution of the 1D KPZ equation under a parabolic rescaling as the variance of the noise variables goes to zero in the intermediate disorder regime, assuming the dependency of neighboring heights is equivariant, symmetric, and at least six times differentiable in a neighborhood of zero. In this paper, we obtained the same convergence result for the half-space model with a Neumann boundary condition.