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March 2017 Random curves, scaling limits and Loewner evolutions
Antti Kemppainen, Stanislav Smirnov
Ann. Probab. 45(2): 698-779 (March 2017). DOI: 10.1214/15-AOP1074

Abstract

In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm’s SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally-invariant scaling limit seems sufficient to deduce the required condition.

Our paper serves as an important step in establishing the convergence of Ising and FK Ising interfaces to SLE curves; moreover, the setup is adapted to branching interface trees, conjecturally describing the full interface picture by a collection of branching SLEs.

Citation

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Antti Kemppainen. Stanislav Smirnov. "Random curves, scaling limits and Loewner evolutions." Ann. Probab. 45 (2) 698 - 779, March 2017. https://doi.org/10.1214/15-AOP1074

Information

Received: 1 October 2013; Revised: 1 June 2015; Published: March 2017
First available in Project Euclid: 31 March 2017

zbMATH: 06797079
MathSciNet: MR3630286
Digital Object Identifier: 10.1214/15-AOP1074

Subjects:
Primary: 60D05
Secondary: 30C35, 60K35, 82B20, 82B31

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 2 • March 2017
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