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Imaginary geometry. I: Interacting SLEs. (English) Zbl 1336.60162
Summary: Fix constants \(\chi >0\) and \(\theta \in [0,2\pi )\), and let \(h\) be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field \(e^{i(h/\chi +\theta )}\) starting at a fixed boundary point of the domain. Letting \(\theta \) vary, one obtains a family of curves that look locally like \(\mathrm{SLE}_\kappa \) processes with \(\kappa \in (0,4)\) (where \(\chi = \tfrac{2}{\sqrt{\kappa }} -\tfrac{ \sqrt{\kappa }}{2}\)), 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 (\(\mathrm {SLE}_{16/\kappa}\)) 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 SLE. For example, we prove that \(\mathrm{SLE}_\kappa (\rho )\) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general \(\mathrm{SLE}_{16/\kappa }(\rho )\) processes.

MSC:
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G60 Random fields
60G15 Gaussian processes
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References:
[1] Ben Arous, G; Deuschel, J-D, The construction of the \((d+1)\)-dimensional Gaussian droplet, Commun. Math. Phys., 179, 467-488, (1996) · Zbl 0858.60096
[2] Camia, F., Newman, C.M.: Two-dimensional critical percolation: the full scaling limit. Commun. Math. Phys. 268(1), 1-38 (2006). arXiv:math/0605035 · Zbl 1117.60086
[3] Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515-580 (2012). arXiv:0910.2045 [math] · Zbl 1257.82020
[4] Duplantier, B; Sheffield, S, Liouville quantum gravity and KPZ, Invent. Math., 185, 333-393, (2011) · Zbl 1226.81241
[5] Dubédat, J, Duality of schramm-Loewner evolutions, Ann. Sci. Éc. Norm. Supér. (4), 42, 697-724, (2009) · Zbl 1205.60147
[6] Dubédat, J, SLE and the free field: partition functions and couplings, J. Am. Math. Soc., 22, 995-1054, (2009) · Zbl 1204.60079
[7] Garban, C; Rohde, S; Schramm, O, Continuity of the SLE trace in simply connected domains, Isr. J. Math., 187, 23-36, (2012) · Zbl 1261.60079
[8] 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
[9] Hu, X; Miller, J; Peres, Y, Thick points of the Gaussian free field, Ann. Probab., 38, 896-926, (2010) · Zbl 1201.60047
[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(3), 1128-1137 (2001). arXiv:math/0002027 · Zbl 1034.82021
[12] Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. In: Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991) · Zbl 0734.60060
[13] Kemppainen, A., Sheffield, S., Schramm, S.: TBD
[14] Lawler, G.F.: Conformally invariant processes in the plane. In: Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005) · Zbl 1074.60002
[15] Lawler, G., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917-955 (2003). arXiv:math/0209343 · Zbl 1030.60096
[16] Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939-995 (2004). arXiv:math/0112234 · Zbl 1126.82011
[17] Miller, J.: Universality for SLE(4) (2010). arXiv:1010.1356 [math] · Zbl 1193.82027
[18] Miller, J, Fluctuations for the Ginzburg-Landau \(∇ ϕ \) interface model on a bounded domain, Commun. Math. Phys., 308, 591-639, (2011) · Zbl 1237.82030
[19] Makarov, N., Smirnov, S.: Off-critical lattice models and massive SLEs. In: XVIth International Congress on Mathematical Physics, pp. 362-371. World Sci. Publ., Hackensack (2010). arXiv:0909.5377 [math] · Zbl 1205.82055
[20] Miller, J., Sheffield, S.: Imaginary geometry II: reversibility of \(\text{ SLE }_κ (ρ _1;ρ _2)\) for \(κ ∈ (0,4)\). To appear in Annal. Probab. (2012). arXiv:1201.1497 [math] · Zbl 1170.60008
[21] Miller, J., Sheffield, S.: Imaginary geometry III: reversibility of \(\text{ SLE }_κ \) for \(κ ∈ (4,8)\). To appear in Annal. Math. (2012). arXiv:1201.1498 [math] · Zbl 1200.60071
[22] Miller, J., Sheffield, S.: Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013). arXiv:1302.4738 [math] · Zbl 1378.60108
[23] Naddaf, A; Spencer, T, On homogenization and scaling limit of some gradient perturbations of a massless free field, Commun. Math. Phys., 183, 55-84, (1997) · Zbl 0871.35010
[24] Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. (2) 161(2), 883-924 (2005). arXiv:math/0106036 · Zbl 1081.60069
[25] Rider, B., Virág, B.: The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN (2) Art. ID rnm006, 33 (2007). arXiv:math/0606663 · Zbl 1261.60079
[26] Revuz, D., Yor, M.: Continuous martingales and Brownian motion. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006
[27] Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221-288 (2000). arXiv:math/9904022 · Zbl 0968.60093
[28] Sheffield, S.: Local sets of the Gaussian free field: slides and audio. http://www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield1/. http://www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield2/. http://www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield3/
[29] Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139(3-4), 521-541 (2007). arXiv:math/0312099 · Zbl 1132.60072
[30] Sheffield, S, Exploration trees and conformal loop ensembles, Duke Math. J., 147, 79-129, (2009) · Zbl 1170.60008
[31] Sheffield, S.: Conformal weldings of random surfaces: SLE and the quantum gravity zipper. To appear in Annal. Probab. (2010). arXiv:1012.4797 [math] · Zbl 1388.60144
[32] Sheffield, S.: Quantum gravity and inventory accumulation. To appear in Annal. Probab. (2011). arXiv:1108.2241 [math]
[33] Smirnov, S, Critical percolation in the plane: conformal invariance, cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math., 333, 239-244, (2001) · Zbl 0985.60090
[34] Smirnov, S, Conformal invariance in random cluster models. I. holomorphic fermions in the Ising model, Ann. Math. (2), 172, 1435-1467, (2010) · Zbl 1200.82011
[35] Schramm, O; Sheffield, S, Harmonic explorer and its convergence to \({\text{ SLE }}_4\), Ann. Probab., 33, 2127-2148, (2005) · Zbl 1095.60007
[36] Schramm, O; Sheffield, S, Contour lines of the two-dimensional discrete Gaussian free field, Acta Math., 202, 21-137, (2009) · Zbl 1210.60051
[37] Schramm, O; Sheffield, S, A contour line of the continuum Gaussian free field, Probab. Theory Relat. Fields, 157, 47-80, (2013) · Zbl 1331.60090
[38] Werner, W.: Random planar curves and Schramm-Loewner evolutions. In: Lectures on Probability Theory and Statistics. Lecture Notes in Math., vol. 1840, pp. 107-195. Springer, Berlin (2004). arXiv:math/0303354 · Zbl 1057.60078
[39] Zhan, D, Duality of chordal SLE, Invent. Math., 174, 309-353, (2008) · Zbl 1158.60047
[40] Zhan, D, Duality of chordal SLE, II, Ann. Inst. Henri Poincaré Probab. Stat., 46, 740-759, (2010) · Zbl 1200.60071
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