Watersheds are Schramm-Loewner Evolution curves

Abstract

We show that in the continuum limit watersheds dividing drainage basins are Schramm-Loewner Evolution (SLE) curves, being described by one single parameter $\kappa$. Several numerical evaluations are applied to ascertain this. All calculations are consistent with SLE${}_{\kappa }$, with $\kappa =1.734\pm 0.005$, being the only known physical example of an SLE with $\kappa <2$. This lies outside the well-known duality conjecture, bringing up new questions regarding the existence and reversibility of dual models. Furthermore it constitutes a strong indication for conformal invariance in random landscapes and suggests that watersheds likely correspond to a logarithmic Conformal Field Theory (CFT) with central charge $c\approx -7/2$.

5 pages and 4 figures

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