Imaginary Geometry I: Interacting SLEs

TL;DR

This paper develops a comprehensive theory of flow lines derived from the Gaussian free field, revealing their interactions, boundary behaviors, and dualities, and introduces new SLE processes with proven continuity and duality properties.

Contribution

It extends existence, uniqueness, and interaction results for flow lines of the GFF, introduces counterflow lines, and establishes new duality and continuity results for SLE processes.

Findings

Flow lines of different angles cross at most once and may bounce.

Flow lines of the same angle merge into a tree structure.

SLE_/ processes are almost surely continuous even when intersecting the boundary.

Abstract

Fix constants \chi >0 and \theta \in [0,2\pi), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field e^{i(h/\chi+\theta)} starting at a fixed boundary point of the domain. Considering all \theta \in [0,2\pi), one obtains a family of curves that look locally like SLE_\kappa, with \kappa \in (0,4), where \chi = 2/\kappa^{1/2} - \kappa^{1/2}/2, 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 (SLE_{16/\kappa}) 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 SLE. For example, we prove that SLE_\kappa(\rho) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general SLE_{16/\kappa}(\rho) processes.