The Annals of Probability

Random curves, scaling limits and Loewner evolutions

Antti Kemppainen and Stanislav Smirnov

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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.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 698-779.

Dates
Received: October 2013
Revised: June 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947307

Digital Object Identifier
doi:10.1214/15-AOP1074

Mathematical Reviews number (MathSciNet)
MR3630286

Zentralblatt MATH identifier
06797079

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B31: Stochastic methods 30C35: General theory of conformal mappings

Keywords
Random curve lattice model weak convergence of probability measures Schramm–Loewner evolution Brownian motion conformal mapping

Citation

Kemppainen, Antti; Smirnov, Stanislav. Random curves, scaling limits and Loewner evolutions. Ann. Probab. 45 (2017), no. 2, 698--779. doi:10.1214/15-AOP1074. https://projecteuclid.org/euclid.aop/1490947307


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