An Invariance Principle of 1D KPZ with Robin Boundary Conditions

TL;DR

This paper proves that a discrete 1D random interface with Robin boundary conditions converges to the KPZ equation's Cole-Hopf solution under specific scaling, extending previous full-space results to half-space models.

Contribution

It extends the invariance principle for the 1D KPZ equation to include Robin boundary conditions in half-space models, broadening the scope of convergence results.

Findings

Convergence of the half-space model to the KPZ equation with Robin boundary conditions.

Extension of previous invariance principles to boundary-influenced models.

Validation of the universality of KPZ scaling in half-space settings.

Abstract

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.