# zbMATH — the first resource for mathematics

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 $$\vec{\rho}\,'$$.

##### MSC:
 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
Full Text:
##### References:
 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.