Schramm–Loewner evolution of the accessible perimeter of isoheight lines of correlated landscapes

Abstract

Real landscapes exhibit long-range height–height correlations, which are quantified by the Hurst exponent [Formula: see text]. We give evidence that for negative [Formula: see text], in spite of the long-range nature of correlations, the statistics of the accessible perimeter of isoheight lines is compatible with Schramm–Loewner evolution curves and therefore can be mapped to random walks, their fractal dimension determining the diffusion constant. Analytic results are recovered for [Formula: see text] and [Formula: see text] and a conjecture is proposed for the values in between. By contrast, for positive [Formula: see text], we find that the random walk is not Markovian but strongly correlated in time. Theoretical and practical implications are discussed.