Oceanic coastline and super-universality of percolation clusters

TL;DR

This paper introduces the concept of oceanic coastlines on rough surfaces, linking them to percolation clusters, and finds a universal fractal dimension for these coastlines across different surface roughness levels.

Contribution

It defines oceanic coastlines on rough surfaces and demonstrates their connection to correlated percolation clusters, revealing a super-universal fractal dimension.

Findings

Fractal dimension of oceanic coastlines is approximately 1.896 for H=0.

The dimension matches the analytic value for percolation (91/48).

Suggests super-universality of coastline fractal dimension across roughness exponents.

Abstract

New fractal subset of a rough surface, the ``oceanic coastline'', is defined. For random Gaussian surfaces with negative Hurst exponent $H<0$, ``oceanic coastlines'' are mapped to the percolation clusters of the (correlated) percolation problem. In the case of rough self-affine surfaces ($H \ge 0$), the fractal dimension of the ``oceanic coastline'' $d_c$ is calculated numerically as a function of the roughness exponent $H$ (using a novel technique of minimizing finite-size effects). For H=0, the result $d_c \approx 1.896$ coincides with the analytic value for the percolation problem (91/48), suggesting a super-universality of $d_c$ for correlated percolation problem.