Contour lines of the two-dimensional discrete Gaussian free field
2009/01/01 by Oded Schramm, Scott Sheffield, Scott Sheffield⋆ · 245 citations
Mathematics · #Arc (geometry) #Boundary (topology) #Computer science #Constant (computer programming) #Convergence (economics) #Domain (mathematical analysis) #Field (mathematics) #Gaussian #Gaussian free field #Geometric Analysis and Curvature Flows #Geometry #Interpolation (computer graphics) #Line (geometry) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Pure mathematics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.1007/s11511-009-0034-y
published in Acta Mathematica 202(1), 21-137 (Mittag-Leffler Institute)
openalex publication_date 2009/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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).
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