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Hadamard’s formula and couplings of SLEs with free field
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  • Published: 21 October 2011

Hadamard’s formula and couplings of SLEs with free field

  • Konstantin Izyurov1 &
  • Kalle Kytölä2 

Probability Theory and Related Fields volume 155, pages 35–69 (2013)Cite this article

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Abstract

The relation between level lines of Gaussian free fields (GFF) and SLE4-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard’s formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann–Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE4 is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.

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Authors and Affiliations

  1. Section de Mathématiques, Université de Genève, 1211, Geneva 4, Switzerland

    Konstantin Izyurov

  2. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin katu 2b), 00014, University of Helsinki, Finland

    Kalle Kytölä

Authors
  1. Konstantin Izyurov
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  2. Kalle Kytölä
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Correspondence to Konstantin Izyurov.

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Izyurov, K., Kytölä, K. Hadamard’s formula and couplings of SLEs with free field. Probab. Theory Relat. Fields 155, 35–69 (2013). https://doi.org/10.1007/s00440-011-0391-2

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  • Received: 20 September 2010

  • Revised: 29 September 2011

  • Published: 21 October 2011

  • Issue Date: February 2013

  • DOI: https://doi.org/10.1007/s00440-011-0391-2

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Mathematics Subject Classification (2000)

  • 60J67
  • 60G60
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