Schramm-Loewner Evolution and isoheight lines of correlated landscapes

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 $SLE$.