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 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).
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- May 2006
- DOI:
- 10.48550/arXiv.math/0605337
- arXiv:
- arXiv:math/0605337
- Bibcode:
- 2006math......5337S
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics;
- Mathematics - Complex Variables
- E-Print:
- 132 pages