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Imaginary geometry. IV: Interior rays, whole-plane reversibility, and space-filling trees. (English) Zbl 1378.60108
Summary: We establish existence and uniqueness for Gaussian free field flow lines started at interior points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at boundary points and use Gaussian free field machinery to determine which chordal $$\mathrm{SLE}_\kappa (\rho _1; \rho _2)$$ processes are time-reversible when $$\kappa < 8$$. Here we extend these results to whole-plane $$\mathrm{SLE}_\kappa (\rho )$$ and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by D. Zhan for $$\kappa \in [0,4]$$ [Probab. Theory Relat. Fields 161, No. 3–4, 561–618 (2015; Zbl 1312.60096)]) to all $$\kappa \in [0,8]$$. We also show that the rays of a given angle (with variable starting point) form a space-filling planar tree. Each branch is a form of $$\mathrm{SLE}_\kappa$$ for some $$\kappa \in (0, 4)$$, and the curve that traces the tree in the natural order (hitting $$x$$ before $$y$$ if the branch from $$x$$ is left of the branch from $$y$$) is a space-filling form of $$\mathrm{SLE}_{\kappa^\prime}$$ where $$\kappa^\prime:= 16/\kappa \in (4, \infty )$$. By varying the boundary data we obtain, for each $$\kappa^\prime>4$$, a family of space-filling variants of $$\mathrm{SLE}_{\kappa^\prime}(\rho )$$ whose time reversals belong to the same family. When $$\kappa^\prime \geq 8$$, ordinary $$\mathrm{SLE}_{\kappa^\prime}$$ belongs to this family, and our result shows that its time-reversal is $$\mathrm{SLE}_{\kappa^\prime}(\kappa^\prime/2 - 4; \kappa^\prime/2 - 4)$$. As applications of this theory, we obtain the local finiteness of $$\mathrm{CLE}_{\kappa^\prime}$$, for $$\kappa^\prime \in (4,8)$$, and describe the laws of the boundaries of $$\mathrm{SLE}_{\kappa^\prime}$$ processes stopped at stopping times.
For Parts I–III, see [the authors, ibid. 164, No. 3–4, 553–705 (2016; Zbl 1336.60162); Ann. Probab. 44, No. 3, 1647–1722 (2016; Zbl 1344.60078); Ann. Math. (2) 184, No. 2, 455–486 (2016; Zbl 1393.60092)], respectively.

##### MSC:
 60J67 Stochastic (Schramm-)Loewner evolution (SLE) 60G60 Random fields 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G15 Gaussian processes 60D05 Geometric probability and stochastic geometry
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