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

• 成果类型: Article
• 署名作者: Miller, Jason; Sheffield, Scott
• 署名单位: Microsoft; Massachusetts Institute of Technology (MIT); University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0780-2
• 发表日期: 2017
• 页码: 729-869
• 关键词: erased random-walks; QUANTUM-GRAVITY; sle; Duality

摘要

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 processes are time-reversible when . Here we extend these results to whole-plane and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for ) to all . 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 for some , 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 where . By varying the boundary data we obtain, for each , a family of space-filling variants of whose time reversals belong to the same family. When , ordinary belongs to this family, and our result shows that its time-reversal is . As applications of this theory, we obtain the local finiteness of , for , and describe the laws of the boundaries of processes stopped at stopping times.