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March 2012 Strong path convergence from Loewner driving function convergence
Scott Sheffield, Nike Sun
Ann. Probab. 40(2): 578-610 (March 2012). DOI: 10.1214/10-AOP627

Abstract

We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version applies in particular to random lattice paths that have chordal SLEκ as a scaling limit, with κ < 8 (nonspace-filling).

Existing SLEκ convergence proofs often begin by showing that the Loewner driving functions of these paths (viewed from ∞) converge to Brownian motion. Unfortunately, this is not sufficient, and additional arguments are required to complete the proofs. We show that driving function convergence is sufficient if it can be established for both parametrization directions and a generic observation point.

Citation

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Scott Sheffield. Nike Sun. "Strong path convergence from Loewner driving function convergence." Ann. Probab. 40 (2) 578 - 610, March 2012. https://doi.org/10.1214/10-AOP627

Information

Published: March 2012
First available in Project Euclid: 26 March 2012

zbMATH: 1255.60148
MathSciNet: MR2952085
Digital Object Identifier: 10.1214/10-AOP627

Subjects:
Primary: 60J67
Secondary: 30C35 , 31A15

Keywords: capacity , Loewner driving convergence , Schramm–Loewner evolutions

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 2 • March 2012
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