Contour lines of the two-dimensional discrete Gaussian free field
- First Online:
- Cite this article as:
- Schramm, O. & Sheffield, S. Acta Math (2009) 202: 21. doi:10.1007/s11511-009-0034-y
- 591 Downloads
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).