Contour lines of the two-dimensional discrete Gaussian free field

Details Contour lines of the two-dimensional discrete Gaussian free field Contour lines of the two-dimensional discrete Gaussian free field

2006-05-12

arxiv.org/abs/math/0605337v3

Mathematics

Probability

132 pages; minor revision

Scientific paper

We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant lambda > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain -- with boundary values -lambda on one boundary arc and lambda 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/lambda-1,b/lambda-1), a variant of SLE(4).

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