Imaginary geometry I: interacting SLEs
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440_2016_Article_698.pdf
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Author(s)
Miller, Jason P.
Sheffield, Scott Roger
Date Issued
March 2016
Journal
Probability Theory and Related Fields
Publisher
Springer-Verlag
Citation
Miller, Jason, and Scott Sheffield. "Imaginary geometry I: interacting SLEs." Probability Theory and Related Fields, vol. 164, no. 3, March 2016, pp 553–705.
Version
Final published version
Abstract
Fix constants and θ∈[0,2π), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field e[superscript i(h/χ+θ)] starting at a fixed boundary point of the domain. Letting θ vary, one obtains a family of curves that look locally like SLE[subscript κ] processes with κ∈(0,4) (where χ=2[sqrt]κ−[sqrt]κ2), which we interpret as the rays of a random geometry with purely imaginary curvature. We extend the fundamental existence and uniqueness results about these paths to the case that the paths intersect the boundary. We also show that flow lines of different angles cross each other at most once but (in contrast to what happens when h is smooth) may bounce off of each other after crossing. Flow lines of the same angle started at different points merge into each other upon intersecting, forming a tree structure. We construct so-called counterflow lines (SLE[subscript 16/κ]) within the same geometry using ordered “light cones” of points accessible by angle-restricted trajectories and develop a robust theory of flow and counterflow line interaction. The theory leads to new results about SLE. For example, we prove that SLE[subscript κ](ρ) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general SLE[subscript 16/κ](ρ) processes.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s00440-016-0698-0
