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A contour line of the continuum Gaussian free field. (English) Zbl 1331.60090
The paper deals with the two-dimensional Gaussian free field on a simply connected domain \(D\) with boundary condition \(-\lambda\) on one boundary arc and boundary condition \(\lambda\) on the complementary arc. In previous papers, the authors studied the discrete Gaussian free field which is a random function on a graph that (when defined on increasingly fine lattices) has the Gaussian free field as a scaling limit. They showed that there is a special constant \(\lambda>0\) such that, if the boundary conditions are set to \(\pm\lambda\) on the two boundary arcs, then the zero chordal contour line connecting the two endpoints of these arcs converges in law to the Schramm-Loewner evolution as the lattice size tends to zero, but \(\lambda\) was not determined. In this paper, the exact value of \(\lambda\) is determined and the Gaussian free field is considered for \(\lambda=\sqrt{\pi/8}\). The authors construct the zero level line in two ways: as the limit of the chordal zero contour lines of the projections of the Gaussian free field onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. They also show that, as a function of the two-dimensional Gaussian free field, the zero level line does not change when the Gaussian free field is modified away from the zero level line and derive some general properties of local sets.

MSC:
60G60 Random fields
60G15 Gaussian processes
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
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