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Acta Mathematica

, 202:21 | Cite as

Contour lines of the two-dimensional discrete Gaussian free field

  • Oded Schramm
  • Scott SheffieldEmail author
Article

Abstract

We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4; a/λ - 1, b/λ - 1), a variant of SLE(4).

Keywords

Random Walk Contour Line Simple Path Simple Random Walk Bessel Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Institut Mittag-Leffler 2009

Authors and Affiliations

  1. 1.Theory Group of Microsoft ResearchOne Microsoft WayRedmondUSA
  2. 2.Courant InstituteNew York UniversityNew YorkUSA

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