Universality and conformal non-invariance in self-affine rough surfaces

TL;DR

This study demonstrates that the statistical properties of iso-height lines in various self-affine rough surfaces are universal across different lattice types, but these contours are generally not conformally invariant unless described by Gaussian free field theory.

Contribution

It reveals the universality of roughness exponents and iso-height line properties across different lattice geometries and shows the lack of conformal invariance in most cases.

Findings

Universality of roughness and growth exponents across lattice types.

Statistical properties of iso-height lines are universal.

Contour lines are not conformally invariant except in Gaussian free field theory.

Abstract

We show numerically that the roughness and growth exponents of a wide range of rough surfaces, such as random deposition with relaxation (RDR), ballistic deposition (BD) and restricted solid-on-solid model (RSOS), are independent of the underlying regular (square, triangular, honeycomb) or random (Voronoi) lattices. In addition we show that the universality holds also at the level of statistical properties of the iso-height lines on different lattices. This universality is revealed by calculating the fractal dimension, loop correlation exponent and the length distribution exponent of the individual contours. We also indicate that the hyperscaling relations are valid for the iso-height lines of all the studied Gaussian and non-Gaussian self-affine rough surfaces. Finally using the direct method of Langlands et.al we show that the contour lines of the rough surfaces are not conformally invariant except when we have simple Gaussian free field theory with zero roughness exponent.