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

TL;DR

This paper advances the understanding of Gaussian free field flow lines, establishing their existence, uniqueness, and properties such as reversibility and space-filling behavior, extending previous results to interior points and whole-plane settings.

Contribution

It extends Gaussian free field flow line theory to interior points and whole-plane cases, proving reversibility for all  , and characterizes space-filling trees formed by these rays.

Findings

Established existence and uniqueness of interior flow lines.

Extended reversibility results to all  .

Described space-filling trees and boundary laws for SLE processes.

Abstract

We establish existence and uniqueness for Gaussian free field flow lines started at {\em 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 {\em boundary} points and use Gaussian free field machinery to determine which chordal \SLE_\kappa(\rho_1; \rho_2) processes are time-reversible when \kappa < 8. Here we extend these results to whole-plane \SLE_\kappa(\rho) and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for \kappa \in [0,4]) to all \kappa \in [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_\kappa for some \kappa \in (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 \SLE_{\kappa'} where \kappa':= 16/\kappa \in (4, \infty). By varying the boundary data we obtain, for each \kappa'>4, a family of space-filling variants of \SLE_{\kappa'}(\rho) whose time reversals belong to the same family. When \kappa' \geq 8, ordinary \SLE_{\kappa'} belongs to this family, and our result shows that its time-reversal is \SLE_{\kappa'}(\kappa'/2 - 4; \kappa'/2 - 4). As applications of this theory, we obtain the local finiteness of \CLE_{\kappa'}, for \kappa' \in (4,8), and describe the laws of the boundaries of \SLE_{\kappa'} processes stopped at stopping times.