A contour line of the continuum Gaussian free field

Abstract

Consider an instance h of the Gaussian free field on a simply connected planar domain D with boundary conditions −λ on one boundary arc and λ on the complementary arc, where λ is the special constant √π/8 . We argue that even though h is defined only as a random distribution, and not as a function, it has a well-defined zero level line γ connecting the endpoints of these arcs, and the law of γ is SLE(4) . We construct γ in two ways: as the limit of the chordal zero contour lines of the projections of h onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. We also show that, as a function of h, γ is “local” (it does not change when h is modified away from γ ) and derive some general properties of local sets.