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 with boundary conditions $-λ$ on one boundary arc and $λ$ on the complementary arc, where $λ$ is the special constant $\sqrt{π/8}$. We argue that even though $h$ is defined only as a random distribution, and not as a function, it has a well-defined zero contour line connecting the endpoints of these arcs, whose law is SLE(4). We construct this contour line 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.

44 pages, 1 figure