Classification of (2+1)-Dimensional Growing Surfaces Using Schramm-Loewner Evolution

TL;DR

This paper investigates the conformal invariance of iso-height lines in various 2D surface growth models, providing evidence they belong to the same universality class as the self-avoiding walk, linked to the KPZ equation.

Contribution

The study demonstrates that iso-height lines in multiple growth models are conformally invariant and share the same universality class as SLE$_{8/3}$, connecting discrete models to the KPZ universality class.

Findings

Iso-height lines exhibit conformal invariance.

Models belong to the same universality class as SLE$_{8/3}$.

All models are consistent with the KPZ universality class.

Abstract

Statistical behavior and scaling properties of iso-height lines in three different saturated two-dimensional grown surfaces with controversial universality classes are investigated using ideas from Schramm-Loewner evolution (SLE$_\kappa$). We present some evidence that the iso-height lines in the ballistic deposition (BD), Eden and restricted solid-on-solid (RSOS) models have conformally invariant properties all in the same universality class as the self-avoiding random walk (SAW), equivalently SLE$_{8/3}$. This leads to the conclusion that all these discrete growth models fall into the same universality class as the Kardar-Parisi-Zhang (KPZ) equation in two dimensions.