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2010

A contour line of the continuum Gaussian free field

Schramm, Oded, Sheffield, Scott

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Consider an instance $h$ of the Gaussian free field on a simply connected planar domain with boundary conditions $-\lambda$ on one boundary arc and $\lambda$ on the complementary arc, where $\lambda$ is the special constant $\sqrt{\pi/8}$.

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.

A contour line of the continuum Gaussian free field · Around

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Leonard Gross · 1967
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R. M. Dudley · 1989
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Philip Protter · 1990
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Ch. Pommerenke · 1992
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Svante Janson · 1997
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