A contour line of the continuum Gaussian free field
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Sheffield_A contour line.pdf
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552.86 KB
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Author(s)
Schramm, Oded
Sheffield, Scott Roger
Date Issued
September 2012
Journal
Probability Theory and Related Fields
Publisher
Springer-Verlag
Citation
Schramm, Oded, and Scott Sheffield. “A contour line of the continuum Gaussian free field.” Probability Theory and Related Fields 157, no. 1 2 (October 16, 2013): 47-80.
Version
Original manuscript
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.
Description
Original manuscript August 14, 2010
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/s00440-012-0449-9
