Coastlines violate the Schramm-Loewner Evolution

TL;DR

This study investigates whether coastlines and landscape isoheight lines follow the Schramm-Loewner Evolution (SLE), finding that coastlines violate SLE and revealing relationships between surface roughness and fractal dimensions.

Contribution

The paper introduces a novel algorithm for analyzing isoheight lines, demonstrating that coastlines do not conform to SLE and establishing a link between Hurst exponent and fractal dimension.

Findings

Isoheight line roughness exponent is approximately unity.

Fractal dimension of isoheight lines decreases linearly with Hurst exponent.

Coastlines violate the Schramm-Loewner Evolution (SLE).

Abstract

Mandelbrot's empirical observation that the coast of Britain is fractal has been confirmed by many authors, but it can be described by the Schramm--Loewner Evolution? Since the self-affine surface of our planet has a positive Hurst exponent, one would not expect a priori any critical behavior. Here, we investigate numerically the roughness and fractal dimension of the isoheight lines of real and artificial landscapes. Using a novel algorithm to take into account overhangs, we find that the roughness exponent of isoheight lines is consistent with unity regardless of the Hurst exponent of the rough surface. Moreover, the effective fractal dimension of the iso-height lines decays linearly with the Hurst exponent of the surface. We perform several tests to verify if the complete and accessible perimeters would follow the Schramm--Loewner Evolution and find that the left passage probability test is clearly violated, implying that coastlines violate SLE.