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The Rohde-Schramm theorem via the Gaussian free field. (English) Zbl 1404.60119

Summary: The Rohde-Schramm theorem states that Schramm-Loewner evolution with parameter \(\kappa\) (or \(\mathrm{SLE}_\kappa\) for short) exists as a random curve, almost surely, if \(\kappa \neq 8\). Here we give a new and concise proof of the result, based on the Liouville quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by S. Rohde and O. Schramm [Ann. Math. (2) 161, No. 2, 883–924 (2005; Zbl 1081.60069)], subsequently shown to be optimal by F. J. Viklund and G. F. Lawler [Acta Math. 209, No. 2, 265–322 (2012; Zbl 1271.82007)], which then implies the Rohde-Schramm theorem.

MSC:

60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60J65 Brownian motion
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References:

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