Schramm-Loewner Evolution and isoheight lines of correlated landscapes

TL;DR

This study investigates the statistical properties of isoheight lines in correlated landscapes, revealing that for negative Hurst exponents, these lines follow Schramm-Loewner Evolution (SLE), while for positive exponents, they do not.

Contribution

It demonstrates that isoheight lines in negatively correlated landscapes conform to SLE, extending understanding of landscape geometry and correlations.

Findings

Isoheight lines for H ≤ 0 are compatible with SLE.

Analytic results are recovered for H = -1 and H = 0.

For H > 0, isoheight lines are not Markovian and do not follow SLE.

Abstract

Real landscapes are usually characterized by long-range height-height correlations, which are quantified by the Hurst exponent $H$. We analyze the statistical properties of the isoheight lines for correlated landscapes of $H\in [-1,1]$. We show numerically that, for $H\leq 0$ the statistics of these lines is compatible with $SLE$ and that established analytic results are recovered for $H=-1$ and $H=0$. This result suggests that for negative $H$, in spite of the long-range nature of correlations, the statistics of isolines is fully encoded in a Brownian motion with a single parameter in the continuum limit. By contrast, for positive $H$ we find that the one-dimensional time series encoding the isoheight lines is not Markovian and therefore not consistent with $SLE$.