Gaussian free field light cones and SLEκ (ρ )

Abstract

© Institute of Mathematical Statistics, 2019. Let h be an instance of the GFF. Fix κ ∈ (0, 4) and χ = 2/√κ-√κ/2. Recall that an imaginary geometry ray is a flow line of ei(h/χ+θ) that looks locally like SLEκ. The light cone with parameter θ ∈ [0, π] is the set of points reachable from the origin by a sequence of rays with angles in [-θ/2, θ/2]. It is known that when θ = 0, the light cone looks like SLEκ, and when θ = π it looks like the range of an SLE16/κ counterflow line. We find that when θ ∈ (0, π) the light cones are either fractal carpets with a dense set of holes or space-filling regions with no holes. We show that every nonspace-filling light cone agrees in law with the range of an SLEκ (ρ) process with ρ ∈ (-2-κ/2) ∨ (κ/2-4),-2). Conversely, the range of any such SLEκ (ρ) process agrees in law with a non-space-filling light cone. As a consequence of our analysis, we obtain the first proof that these SLEκ (ρ) processes are a.s. continuous curves and show that they can be constructed as natural path-valued functions of the GFF.