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Contour lines of the two-dimensional discrete Gaussian free field. (English) Zbl 1210.60051
The paper deals with the 2-dimensional massless Gaussian free field (GFF) which is a 2-dimensional-time analog of the Wiener process. The GFF is a scaling limit of several discrete models for random surfaces just as the Wiener process is the hydrodynamic limit of random evolutions. The authors prove that the chordal contour lines of the GFF converge to forms of SLE(4), where SLE(4) is the scaling limit of a random interface called the harmonic explorer. Specifically, there is a constant \(c>0\) such that h is an interpolation of the discrete GFF on a Jordan domain with values \(-c\) on one boundary arc and \(c\) on the complementary arc, the zero level line of \(c\) 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\((4;a/c-1,b/c-1)\), a variant of SLE(4).

MSC:
60G60 Random fields
60G15 Gaussian processes
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