There are No Nice Interfaces in 2+1 Dimensional SOS-Models in Random Media

TL;DR

This paper proves that in disordered media, rigid interfaces in SOS models cannot exist in dimensions two or lower at any temperature, highlighting fundamental differences from higher dimensions.

Contribution

It demonstrates the non-existence of translation covariant Gibbs states for interfaces in disordered SOS models in low dimensions, extending Aizenman and Wehr's theorem.

Findings

Rigid interfaces do not exist in $d\,\leq\,2$ for disordered SOS models.

The proof adapts a theorem of Aizenman and Wehr to this context.

Contrasts with higher-dimensional cases where interfaces can exist.

Abstract

We prove that in dimension $d\leq 2$ translation covariant Gibbs states describing rigid interfaces in a disordered solid-on-solid (SOS) cannot exist for any value of the temperature, in contrast to the situation in $d\geq 3$. The prove relies on an adaptation of a theorem of Aizenman and Wehr. Keywords: Disordered systems, interfaces, SOS-model