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Probability Theory and Related Fields

, Volume 157, Issue 1–2, pp 47–80 | Cite as

A contour line of the continuum Gaussian free field

  • Oded Schramm
  • Scott SheffieldEmail author
Article

Abstract

Consider an instance \(h\) of the Gaussian free field on a simply connected planar domain \(D\) with boundary conditions \(-\lambda \) on one boundary arc and \(\lambda \) on the complementary arc, where \(\lambda \) is the special constant \(\sqrt{\pi /8}\). We argue that even though \(h\) is defined only as a random distribution, and not as a function, it has a well-defined zero level line \(\gamma \) connecting the endpoints of these arcs, and the law of \(\gamma \) is \(\mathrm{SLE}(4)\). We construct \(\gamma \) in two ways: as the limit of the chordal zero contour lines of the projections of \(h\) onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. We also show that, as a function of \(h, \gamma \) is “local” (it does not change when \(h\) is modified away from \(\gamma \)) and derive some general properties of local sets.

Mathematics Subject Classification

60J67 

Notes

Acknowledgments

We wish to thank Vincent Beffara, Richard Kenyon, Jané Kondev, Julien Dubédat, and David Wilson for inspiring and useful conversations. In particular, Kenyon and the first author worked out a coupling of the GFF and \(\mathrm{SLE}(8)\) early on, which is quite similar to the coupling between the GFF and \(\mathrm{SLE}(4)\) presented here. We also thank Jason Miller for helpful comments on a draft of this paper.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

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