Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes

TL;DR

This study demonstrates that the external perimeter of percolation clusters in correlated landscapes with Hurst exponent H between -1 and 0 behaves like SLE curves, revealing Markovian and Gaussian properties.

Contribution

It provides numerical evidence linking correlated landscape perimeters to Schramm-Loewner evolution, extending understanding of fractal boundaries in correlated systems.

Findings

Perimeters follow SLE curves for H in [-1,0]

External perimeters exhibit Markovian properties

Winding angle variance grows logarithmically with size

Abstract

Motivated by the fact that many physical landscapes are characterized by long-range height-height correlations that are quantified by the Hurst exponent H, we investigate the statistical properties of the iso-height lines of correlated surfaces in the framework of Schramm-Loewner evolution (SLE). We show numerically that in the continuum limit the external perimeter of a percolating cluster of correlated surfaces with H between -1 and zero is statistically equivalent to SLE curves. Our results suggest that the external perimeter also retains the Markovian properties, confirmed by the absence of time correlations in the driving function and the fact that the latter is Gaussian distributed for any specific time. We also confirm that for all H the variance of the winding angle grows logarithmically with size.