Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

Abstract

© 2017, The Author(s). We establish existence and uniqueness for Gaussian free field flow lines started at interior points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at boundary points and use Gaussian free field machinery to determine which chordal SLE κ(ρ1; ρ2) processes are time-reversible when κ< 8. Here we extend these results to whole-plane SLE κ(ρ) and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for κ∈ [ 0 , 4 ]) to all κ∈ [ 0 , 8 ]. We also show that the rays of a given angle (with variable starting point) form a space-filling planar tree. Each branch is a form of SLE κ for some κ∈ (0 , 4) , and the curve that traces the tree in the natural order (hitting x before y if the branch from x is left of the branch from y) is a space-filling form of SLEκ′ where κ′: = 16 / κ∈ (4 , ∞). By varying the boundary data we obtain, for each κ′> 4 , a family of space-filling variants of SLE whose time reversals belong to the same family. When κ′≥ 8 , ordinary SLEκ′ belongs to this family, and our result shows that its time-reversal is SLEκ′(κ′/2-4/2-4). As applications of this theory, we obtain the local finiteness of CLEκ′, for κ′∈ (4 , 8) , and describe the laws of the boundaries of SLEκ′ processes stopped at stopping times.