Imaginary geometry I: interacting SLEs

• 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/s00440-016-0698-0
• 发表日期: 2016
• 页码: 553-705
• 关键词: erased random-walks; conformal-invariance; critical percolation; Duality; FORMULA

摘要

Fix constants and , and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field starting at a fixed boundary point of the domain. Letting vary, one obtains a family of curves that look locally like processes with (where ), 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 () 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 . For example, we prove that processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general processes.