Conformal Invariance of Iso-height Lines in two-dimensional KPZ Surface

TL;DR

This paper demonstrates that iso-height lines in the 2D KPZ surface are conformally invariant and can be described by SLE curves, revealing new analytical insights into KPZ surface dynamics.

Contribution

It establishes the conformal invariance of iso-height lines in 2D KPZ surfaces and links them to SLE curves, providing new analytical results for KPZ dynamics.

Findings

Iso-height lines are conformally invariant and equivalent to SLE$_ppa$ with ppa=8/3.

Absence of non-linear term changes ppa from 8/3 to 4, linking EW surface lines to O(2) spin model.

Provides evidence for universality class of iso-height lines in KPZ and EW models.

Abstract

The statistics of the iso-height lines in (2+1)-dimensional Kardar-Parisi-Zhang (KPZ) model is shown to be conformal invariant and equivalent to those of self-avoiding random walks. This leads to a rich variety of new exact analytical results for the KPZ dynamics. We present direct evidence that the iso-height lines can be described by the family of conformal invariant curves called Schramm-Loewner evolution (or $SLE_\kappa$) with diffusivity $\kappa=8/3$. It is shown that the absence of the non-linear term in the KPZ equation will change the diffusivity $\kappa$ from 8/3 to 4, indicating that the iso-height lines of the Edwards-Wilkinson (EW) surface are also conformally invariant, and belong to the universality class of the domain walls in the O(2) spin model.