# Duality of chordal SLE, II

Annales de l'I.H.P. Probabilités et statistiques (2010)

- Volume: 46, Issue: 3, page 740-759
- ISSN: 0246-0203

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topZhan, Dapeng. "Duality of chordal SLE, II." Annales de l'I.H.P. Probabilités et statistiques 46.3 (2010): 740-759. <http://eudml.org/doc/240299>.

@article{Zhan2010,

abstract = {We improve the geometric properties of processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for κ∈(4, 8), the boundary of a standard chordal SLE(κ) hull stopped on swallowing a fixed x∈ℝ∖\{0\} is the image of some trace started from a random point. Using this fact together with a similar proposition in the case that κ≥8, we obtain a description of the boundary of a standard chordal SLE(κ) hull for κ>4, at a finite stopping time. Finally, we prove that for κ>4, in many cases, a chordal or strip trace a.s. ends at a single point.},

author = {Zhan, Dapeng},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {SLE; duality; coupling technique},

language = {eng},

number = {3},

pages = {740-759},

publisher = {Gauthier-Villars},

title = {Duality of chordal SLE, II},

url = {http://eudml.org/doc/240299},

volume = {46},

year = {2010},

}

TY - JOUR

AU - Zhan, Dapeng

TI - Duality of chordal SLE, II

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2010

PB - Gauthier-Villars

VL - 46

IS - 3

SP - 740

EP - 759

AB - We improve the geometric properties of processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for κ∈(4, 8), the boundary of a standard chordal SLE(κ) hull stopped on swallowing a fixed x∈ℝ∖{0} is the image of some trace started from a random point. Using this fact together with a similar proposition in the case that κ≥8, we obtain a description of the boundary of a standard chordal SLE(κ) hull for κ>4, at a finite stopping time. Finally, we prove that for κ>4, in many cases, a chordal or strip trace a.s. ends at a single point.

LA - eng

KW - SLE; duality; coupling technique

UR - http://eudml.org/doc/240299

ER -

## References

top- [1] L. V. Ahlfors. Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York, 1973. Zbl0272.30012MR357743
- [2] V. Beffara. Hausdorff dimensions for SLE6. Ann. Probab. 32 (2004) 2606–2629. Zbl1055.60036MR2078552
- [3] V. Beffara. The dimension of the SLE curves. Ann. Probab. 36 (2008) 1421–1452. Zbl1165.60007MR2435854
- [4] J. Dubédat. Duality of Schramm–Loewner evolutions. Ann. Sci. École Norm. Sup. (4) 42 (2009) 697–724. Zbl1205.60147MR2571956
- [5] G. F. Lawler, O. Schramm and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004) 939–995. Zbl1126.82011MR2044671
- [6] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1991. Zbl0917.60006MR1083357
- [7] S. Rohde and O. Schramm. Basic properties of SLE. Ann. Math. 161 (2005) 883–924. Zbl1081.60069MR2153402
- [8] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000) 221–288. Zbl0968.60093MR1776084
- [9] D. Zhan. Duality of chordal SLE. Inven. Math. 174 (2008) 309–353. Zbl1158.60047MR2439609
- [10] D. Zhan. The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36 (2008) 467–529. Zbl1153.60057MR2393989
- [11] D. Zhan. Reversibility of chordal SLE. Ann. Probab. 36 (2008) 1472–1494. Zbl1157.60051MR2435856

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