Imaginary geometry I: interacting SLEs

Abstract

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