A contour line of the continuum Gaussian free field

By Oded Schramm, Scott Sheffield. 2010

205 citations recorded by the source corpus.

Consider an instance $$h$$ of the Gaussian free field on a simply connected planar domain $$D$$ 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 level line $$\gamma $$ connecting the endpoints of these arcs, and the law of $$\gamma $$ is $$\mathrm{SLE}(4)$$. We construct $$\gamma $$ 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 $$h, \gamma $$ is “local” (it does not change when $$h$$ is modified away from $$\gamma $$) and derive some general properties of local sets.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bea71fd8f1d9483aa984160b1cbffee3b89933ff

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “The notion of a local set, first introduced in [21] for ordinary GFFs, serves to generalize the Markov property to the setting in which we condition the GFF on its values on a random (rather than deterministic) closed set.”