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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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##### References:
 [1] Aldous, D, The continuum random tree. I, Ann. Probab., 19, 1-28, (1991) · Zbl 0722.60013 [2] Aldous, D.: The continuum random tree. II. An overview. In: Stochastic Analysis (Durham, 1990), Volume 167 of London Mathematical Society Lecture Note Series, pp. 23-70. Cambridge University Press, Cambridge (1991) · Zbl 0791.60008 [3] Aldous, D, The continuum random tree. III, Ann. Probab., 21, 248-289, (1993) · Zbl 0791.60009 [4] Beffara, V, The dimension of the SLE curves, Ann. Probab., 36, 1421-1452, (2008) · Zbl 1165.60007 [5] Duplantier, B., Miller, J., Sheffield, S.: Liouville quantum gravity as a mating of trees. ArXiv e-prints, September 2014. arXiv:1409.7055 · Zbl 1170.60008 [6] Duplantier, B; Sheffield, S, Liouville quantum gravity and KPZ, Invent. Math., 185, 333-393, (2011) · Zbl 1226.81241 [7] Dubédat, J, Duality of schramm-Loewner evolutions, Ann. Sci. Éc. Norm. Supér. (4), 42, 697-724, (2009) · Zbl 1205.60147 [8] Dubédat, J, SLE and the free field: partition functions and couplings, J. Am. Math. Soc., 22, 995-1054, (2009) · Zbl 1204.60079 [9] Hagendorf, C; Bernard, D; Bauer, M, The Gaussian free field and $${{\rm SLE}}_4$$ on doubly connected domains, J. Stat. Phys., 140, 1-26, (2010) · Zbl 1193.82027 [10] Izyurov, K; Kytölä, K, Hadamard’s formula and couplings of SLEs with free field, Probab. Theory Relat. Fields, 155, 35-69, (2013) · Zbl 1269.60067 [11] Kenyon, R, Dominos and the Gaussian free field, Ann. Probab., 29, 1128-1137, (2001) · Zbl 1034.82021 [12] Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, Volume 113 of Graduate Texts in Mathematics. Springer, New York (1991) · Zbl 0734.60060 [13] Lawler, G.F.: Conformally Invariant Processes in the Plane, Volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2005) [14] Lawler, G.F.: Continuity of radial and two-sided radial $${{\rm SLE}}_κ$$ at the terminal point. ArXiv e-prints, April 2011. arXiv:1104.1620 · Zbl 1081.60069 [15] Lawler, GF; Schramm, O; Werner, W, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab., 32, 939-995, (2004) · Zbl 1126.82011 [16] Makarov, N., Smirnov, S.: Off-critical lattice models and massive SLEs. In: XVIth International Congress on Mathematical Physics, pp. 362-371. World Science Publisher, Hackensack (2010). arXiv:0909.5377 · Zbl 1205.82055 [17] Miller, J., Sheffield, S.: An axiomatic characterization of the Brownian map. ArXiv e-prints, June 2015. arXiv:1506.03806 · Zbl 0968.60093 [18] Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric. ArXiv e-prints, July 2015. arXiv:1507.00719 · Zbl 1269.60067 [19] Miller, J., Sheffield, S.: Liouville quantum gravity spheres as matings of finite-diameter trees. ArXiv e-prints, June 2015. arXiv:1506.03804 · Zbl 1408.60074 [20] Miller, J., Sheffield, S.: Gaussian free field light cones and SLE$$_κ (ρ )$$. ArXiv e-prints, June 2016. arXiv:1606.02260 · Zbl 1132.60072 [21] Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. ArXiv e-prints, May 2016. arXiv:1605.03563 [22] Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map III: the conformal structure is determined. ArXiv e-prints, August 2016. arXiv:1608.05391 · Zbl 1344.60078 [23] Miller, J; Sheffield, S, Imaginary geometry I: interacting sles, Probab. Theory Relat. Fields, 164, 553-705, (2016) · Zbl 1336.60162 [24] Miller, J; Sheffield, S, Imaginary geometry II: reversibility of $${{\rm SLE}}_κ (ρ _1;ρ _2)$$ for $$κ ∈ (0,4)$$, Ann. Probab., 44, 1647-1722, (2016) · Zbl 1344.60078 [25] Miller, J; Sheffield, S, Imaginary geometry III: reversibility of $${{\rm SLE}}_κ$$ for $$κ ∈ (4,8)$$, Ann. Math. (2), 184, 455-486, (2016) · Zbl 1393.60092 [26] Miller, J; Sheffield, S, Quantum Loewner evolution, Duke Math. J., 165, 3241-3378, (2016) · Zbl 1364.82023 [27] Miller, J., Sheffield, S., Werner, W.: CLE percolations. ArXiv e-prints, February 2016. arXiv:1602.03884 · Zbl 1193.82027 [28] Miller, J; Wu, H, Intersections of SLE paths: the double and cut point dimension of SLE, Probab. Theory Relat. Fields, 167, 45-105, (2017) · Zbl 1408.60074 [29] Rohde, S., Schramm, O.: Unpublished · Zbl 1205.82063 [30] Rohde, S; Schramm, O, Basic properties of SLE, Ann. Math. (2), 161, 883-924, (2005) · Zbl 1081.60069 [31] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006 [32] Schramm, O, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math., 118, 221-288, (2000) · Zbl 0968.60093 [33] Sheffield, S.: Local sets of the Gaussian free field: slides and audio. www.fields.utoronto.ca/0506/percolationsle/sheffield1, www.fields.utoronto.ca/audio/0506/percolationsle/sheffield2, www.fields.utoronto.ca/audio/0506/percolationsle/sheffield3 · Zbl 1331.60090 [34] Sheffield, S, Gaussian free fields for mathematicians, Probab. Theory Relat. Fields, 139, 521-541, (2007) · Zbl 1132.60072 [35] Sheffield, S, Exploration trees and conformal loop ensembles, Duke Math. J., 147, 79-129, (2009) · Zbl 1170.60008 [36] Sheffield, S, Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab., 44, 3474-3545, (2016) · Zbl 1388.60144 [37] Sheffield, S, Quantum gravity and inventory accumulation, Ann. Probab., 44, 3804-3848, (2016) · Zbl 1359.60120 [38] Schramm, O; Sheffield, S, A contour line of the continuum Gaussian free field, Probab. Theory Relat. Fields, 157, 47-80, (2013) · Zbl 1331.60090 [39] Schramm, O., Wilson, D.B.: Private communication · Zbl 1205.60147 [40] Schramm, O; Wilson, DB, SLE coordinate changes, N. Y. J. Math., 11, 659-669, (2005) · Zbl 1094.82007 [41] Sheffield, S; Werner, W, Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Ann. Math. (2), 176, 1827-1917, (2012) · Zbl 1271.60090 [42] Werner, W.: Random planar curves and Schramm-Loewner evolutions. In: Lectures on Probability Theory and Statistics, Volume 1840 of Lecture Notes in Mathematics, pp. 107-195. Springer, Berlin (2004). arXiv:math/0303354 · Zbl 1057.60078 [43] Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pp. 296-303. ACM, New York (1996) · Zbl 0946.60070 [44] Zhan, D, Duality of chordal SLE, Invent. Math., 174, 309-353, (2008) · Zbl 1158.60047 [45] Zhan, D, Reversibility of chordal SLE, Ann. Probab., 36, 1472-1494, (2008) · Zbl 1157.60051 [46] Zhan, D, Duality of chordal SLE, II, Ann. Inst. Henri Poincaré Probab. Stat., 46, 740-759, (2010) · Zbl 1200.60071 [47] Zhan, D, Reversibility of some chordal $${{\rm SLE}}(κ ; ρ )$$ traces, J. Stat. Phys., 139, 1013-1032, (2010) · Zbl 1205.82063 [48] Zhan, D, Reversibility of whole-plane SLE, Probab. Theory Relat. Fields, 161, 561-618, (2015) · Zbl 1312.60096
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