Contour lines of the two-dimensional discrete Gaussian free field

By Oded Schramm, Scott Sheffield. 2006

325 citations recorded by the source corpus.

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).

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7003ee7420bff12d2eac9b75168a2901c73cbf22

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Universality in the 2D Ising model and conformal invariance of fermionic observables (2009, methodology): “Situation is somewhat easier with the observables for the Harmonic Explorer and Discrete Gaussian Free Field, as discussed by Schramm and Sheffield [SS05, SS09] – both are harmonic and solving Dirichlet problem with bounded boundary values, so the generalization from the original triangular lattice ”
Imaginary geometry I: interacting SLEs (2012, background): “This has not been proved, but an analogous result has been shown for level sets of h [36,37].”
Two-Dimensional Critical Percolation: The Full Scaling Limit (2006, background): “We conclude this section by remarking that the Continuum Nonsimple Loop process is just one example of a family of “conformal loop ensembles” that are related to SLE and to the Gaussian Free Field (see [ 33 , 35, 42, 44, 45]), and are conjectured to describe the full scaling limit of statistical mec”
Duality of chordal SLE (2007, background): “…to 0, e.g., the convergence of critical percolation on triangular lattice to SLE(6) [16], loop-erased random walk (LERW) to SLE(2) [9,18], uniform spanning tree (UST) Peano curve to SLE(8) [9], Gaussian free field contour line to SLE(4) [13], and some Ising models to SLE(3) and SLE(16/3) [15].”
A contour line of the continuum Gaussian free field (2010, methodology): “The authors showed in [SS09] that a certain level line of the DGFF has SLE(4) as a scaling limit.”