Imaginary geometry I: interacting SLEs

By Jason Miller, Scott Sheffield. 2012

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Fix constants $$\chi >0$$χ>0 and $$\theta \in [0,2\pi )$$θ∈[0,2π), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field $$e^{i(h/\chi +\theta )}$$ei(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 $$\hbox {SLE}_\kappa $$SLEκ processes with $$\kappa \in (0,4)$$κ∈(0,4) (where $$\chi = \tfrac{2}{\sqrt{\kappa }} -\tfrac{ \sqrt{\kappa }}{2}$$χ=2κ-κ2), 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 ($$\hbox {SLE}_{16/\kappa }$$SLE16/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 $$\hbox {SLE}_\kappa (\rho )$$SLEκ(ρ) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general $$\hbox {SLE}_{16/\kappa }(\rho )$$SLE16/κ(ρ) processes.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dcc88fc5fbfd1462ff0f16c6848ae017e1c30c95

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Citations in the corpus

Imaginary geometry II: Reversibility of SLEκ(ρ1;ρ2) for κ∈(0,4). (2016, methodology): “The latter provides a mechanism for constructing couplings of many SLEκ(ρ ; ρ) strands in such a way that it is easy to compute the conditional law of one of the curves given the realization of the others [MS12a].”
Conformal weldings of random surfaces: SLE and the quantum gravity zipper (2010, background): “3) this a major result due to Dubédat [Dub09] (see also the exposition on this point in [MS12a]).”
Liouville quantum gravity as a mating of trees (2014, methodology): “Using the version of SLE duality established in [MS12a] and Proposition 6.”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “This step is carried out in some generality in [3, 26, 14].”