Gaussian free field light cones and SLEκ (ρ )
Name
1606.02260.pdf
Description
Submitted version
Size
724.79 KB
Format
Adobe PDF
Checksum (MD5)
6aed09ee4b48f363ba109b4bc00d98df
Author(s)
Miller, Jason
Sheffield, Scott
Date Issued
2019
Journal
The Annals of Probability
Publisher
Institute of Mathematical Statistics
Version
Original manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1214/18-AOP1331
