2013/02/19 by Jason Miller, Scott Sheffield⋆, Miller, Jason +1 · 6 citations
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1302.4738
openalex publication_date 2013/02/19 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We establish existence and uniqueness for Gaussian free field flow lines\nstarted at em interior points of a planar domain. We interpret these as rays\nof a random geometry with imaginary curvature and describe the way distinct\nrays intersect each other and the boundary.\n Previous works in this series treat rays started at em boundary points and\nuse Gaussian free field machinery to determine which chordal\n SLE_\κ(\ρ1; \ρ2) processes are time-reversible when \κ < 8. Here\nwe extend these results to whole-plane SLE_\κ(\ρ) and establish\ncontinuity and transience of these paths. In particular, we extend ordinary\nwhole-plane SLE reversibility (established by Zhan for \κ \∈ [0,4]) to all\n\κ \∈ [0,8].\n We also show that the rays of a given angle (with variable starting point)\nform a space-filling planar tree. Each branch is a form of SLE_\κ for some\n\κ \∈ (0, 4), and the curve that traces the tree in the natural order\n(hitting x before y if the branch from x is left of the branch from y) is a\nspace-filling form of SLE\κ' where \κ':= 16/\κ \∈ (4, \∞).\nBy varying the boundary data we obtain, for each \κ'>4, a family of\nspace-filling variants of SLE\κ'(\ρ) whose time reversals belong to\nthe same family. When \κ' \≥ 8, ordinary SLE\κ' belongs to this\nfamily, and our result shows that its time-reversal is SLE\κ'(\κ'/2\n- 4; \κ'/2 - 4).\n As applications of this theory, we obtain the local finiteness of\n CLE\κ', for \κ' \∈ (4,8), and describe the laws of the boundaries\nof SLE\κ' processes stopped at stopping times.\n