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Duality of chordal SLE. (English) Zbl 1158.60047
Author’s summary: We derive some geometric properties of chordal \(\text{SLE}(\kappa ;\vec{\rho})\) processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal \(\text{SLE}(\kappa ;\vec{\rho})\) process has the same distribution as the image of a chordal \(\text{SLE}(\kappa';\vec{\rho}\,'\)) trace, where \(\kappa >4, \kappa'=16/\kappa \), and the forces \(\vec{\rho}\) and \(\vec{\rho}\,'\) are suitably chosen. We find that for \(\kappa \geq 8\), the boundary of a standard chordal \(\text{SLE}(\kappa )\) hull stopped on swallowing a fixed \(x\in\mathbb{R}\setminus\{0\}\) is the image of some \(\text{SLE}(16/\kappa ;\vec{\rho})\) trace started from \(x\). Then we obtain a new proof of the fact that chordal \(\text{SLE}(\kappa )\) trace is not reversible for \(\kappa >8\). We also prove that the reversal of \(\text{SLE}(4;\vec{\rho})\) trace has the same distribution as the time-change of some \(\text{SLE}(4;\vec{\rho}\,')\) trace for certain values of \(\vec{\rho}\) and \(\vec{\rho}\,'\).

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G99 Stochastic processes
60J65 Brownian motion
30C35 General theory of conformal mappings
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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[1] Ahlfors, L.V.: Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York (1973) · Zbl 0272.30012
[2] Bauer, M., Bernard, D., Houdayer, J.: Dipolar stochastic Loewner evolutions. J. Stat. Mech. P03001 (2005)
[3] Beffara, V.: The dimension of the SLE curves. Ann. Probab. 36(4), 1421–1452 (2008) · Zbl 1165.60007
[4] Dubédat, J.: SLE({\(\kappa\)},{\(\rho\)}) martingales and duality. Ann. Probab. 33(1), 223–243 (2005) · Zbl 1096.60037
[5] Dubédat, J.: Commutation relations for SLE. Commun. Pure Appl. Math. 60(12), 1792–1847 (2007) · Zbl 1137.82009
[6] Lawler, G.F.: Conformally Invariant Processes in the Plane. Am. Math. Soc., Providence, RI (2005) · Zbl 1074.60002
[7] Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187(2), 237–273 (2001) · Zbl 1005.60097
[8] Lawler, G.F., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917–955 (2003) · Zbl 1030.60096
[9] 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) · Zbl 1126.82011
[10] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991) · Zbl 0731.60002
[11] Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161(2), 883–924 (2005) · Zbl 1081.60069
[12] Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000) · Zbl 0968.60093
[13] Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Probab. Theory Relat. Fields (to appear), arXiv:math.PR/0605337 · Zbl 1210.60051
[14] Schramm, O., Wilson, D.B.: SLE coordinate changes. New York J. Math. 11, 659–669 (2005) · Zbl 1094.82007
[15] Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians, vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich (2006) · Zbl 1112.82014
[16] Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci., Paris, Sér. I, Math. 333(3), 239–244 (2001) · Zbl 0985.60090
[17] Wilson, D.B.: Generating random trees more quickly than the cover time. In: Proceedings of the 28th ACM Symposium on the Theory of Computing, pp. 296–303. Assoc. Comput. Mach., New York (1996) · Zbl 0946.60070
[18] Zhan, D.: The Scaling Limits of Planar LERW in finitely connected domains. Ann. Probab. 36(2), 467–529 (2008) · Zbl 1153.60057
[19] Zhan, D.: Reversibility of chordal SLE. Ann. Probab. 36(4), 1472–1494 (2008) · Zbl 1157.60051
[20] Zhan, D.: Random Loewner chains in Riemann surfaces. PhD thesis, Caltech (2004)
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