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Contour lines of the two-dimensional discrete Gaussian free field

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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Correspondence to Scott Sheffield.

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During the revision process of this article, Oded Schramm unexpectedly died. I am deeply indebted for all I learned working with him, for his profound personal warmth, for his legendary vision and skill. There was never a better colleague, never a better friend. He will be dearly missed. (Scott Sheffield)

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Schramm, O., Sheffield, S. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math 202, 21 (2009). https://doi.org/10.1007/s11511-009-0034-y

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  • DOI: https://doi.org/10.1007/s11511-009-0034-y

Keywords

  • Random Walk
  • Contour Line
  • Simple Path
  • Simple Random Walk
  • Bessel Process