Acta Mathematica

, 202:21 | Cite as

Contour lines of the two-dimensional discrete Gaussian free field

  • Oded Schramm
  • Scott SheffieldEmail author


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).


Random Walk Contour Line Simple Path Simple Random Walk Bessel Process 
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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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