Acta Mathematica

Contour lines of the two-dimensional discrete Gaussian free field

Oded Schramm and Scott Sheffield

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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).


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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Acta Math., Volume 202, Number 1 (2009), 21-137.

Received: 18 October 2006
First available in Project Euclid: 31 January 2017

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2009 © Institut Mittag-Leffler


Schramm, Oded; Sheffield, Scott. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 (2009), no. 1, 21--137. doi:10.1007/s11511-009-0034-y.

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