Duality of chordal SLE, II
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 740-759.

Nous améliorons des résultats précédemment obtenus concernant les propriétés géométriques des processus , que nous utilisons ensuite pour étudier la propriété dite de dualité des processus SLE. Nous prouvons que pour κ∈(4, 8), la frontière de l'enveloppe d'un SLE(κ) chordal standard arrêté quand il disconnecte un point fixe x∈ℝ\{0} de l'infini est une courbe issue d'un point aléatoire. Nous obtenons ainsi une description de la frontière de l'enveloppe d'un SLE(κ) pour κ>4. Finalement, nous démontrons que pour κ>4, dans de nombreux cas, la courbe de processus généralisés (par exemple dans une bande) se termine presque sûrement en un point unique.

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.

DOI : 10.1214/09-AIHP340
Classification : 30C20, 60H05
Mots-clés : SLE, duality, coupling technique
@article{AIHPB_2010__46_3_740_0,
     author = {Zhan, Dapeng},
     title = {Duality of chordal {SLE,} {II}},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {740--759},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {3},
     year = {2010},
     doi = {10.1214/09-AIHP340},
     mrnumber = {2682265},
     zbl = {1200.60071},
     language = {en},
     url = {https://www.numdam.org/articles/10.1214/09-AIHP340/}
}
TY  - JOUR
AU  - Zhan, Dapeng
TI  - Duality of chordal SLE, II
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2010
SP  - 740
EP  - 759
VL  - 46
IS  - 3
PB  - Gauthier-Villars
UR  - https://www.numdam.org/articles/10.1214/09-AIHP340/
DO  - 10.1214/09-AIHP340
LA  - en
ID  - AIHPB_2010__46_3_740_0
ER  - 
%0 Journal Article
%A Zhan, Dapeng
%T Duality of chordal SLE, II
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2010
%P 740-759
%V 46
%N 3
%I Gauthier-Villars
%U https://www.numdam.org/articles/10.1214/09-AIHP340/
%R 10.1214/09-AIHP340
%G en
%F AIHPB_2010__46_3_740_0
Zhan, Dapeng. Duality of chordal SLE, II. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 740-759. doi : 10.1214/09-AIHP340. https://www.numdam.org/articles/10.1214/09-AIHP340/

[1] L. V. Ahlfors. Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York, 1973. | MR | Zbl

[2] V. Beffara. Hausdorff dimensions for SLE6. Ann. Probab. 32 (2004) 2606-2629. | MR | Zbl

[3] V. Beffara. The dimension of the SLE curves. Ann. Probab. 36 (2008) 1421-1452. | MR | Zbl

[4] J. Dubédat. Duality of Schramm-Loewner evolutions. Ann. Sci. École Norm. Sup. (4) 42 (2009) 697-724. | Numdam | MR | Zbl

[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. | MR | Zbl

[6] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1991. | MR | Zbl

[7] S. Rohde and O. Schramm. Basic properties of SLE. Ann. Math. 161 (2005) 883-924. | MR | Zbl

[8] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000) 221-288. | MR | Zbl

[9] D. Zhan. Duality of chordal SLE. Inven. Math. 174 (2008) 309-353. | MR | Zbl

[10] D. Zhan. The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36 (2008) 467-529. | MR | Zbl

[11] D. Zhan. Reversibility of chordal SLE. Ann. Probab. 36 (2008) 1472-1494. | MR | Zbl

  • Gwynne, Ewain; Pfeffer, Joshua; Park, Minjae Loewner evolution driven by complex Brownian motion, The Annals of Probability, Volume 51 (2023) no. 6 | DOI:10.1214/23-aop1639
  • Yearwood, Stephen The topology of SLEκ is random for κ>4, Electronic Journal of Probability, Volume 27 (2022) no. none | DOI:10.1214/22-ejp871
  • Katori, Makoto; Koshida, Shinji Three phases of multiple SLE driven by non-colliding Dyson’s Brownian motions, Journal of Physics A: Mathematical and Theoretical, Volume 54 (2021) no. 32, p. 325002 | DOI:10.1088/1751-8121/ac0dee
  • Gwynne, Ewain; Pfeffer, Joshua External diffusion-limited aggregation on a spanning-tree-weighted random planar map, The Annals of Probability, Volume 49 (2021) no. 4 | DOI:10.1214/20-aop1486
  • Ding, Jian; Gwynne, Ewain The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds, Communications in Mathematical Physics, Volume 374 (2020) no. 3, p. 1877 | DOI:10.1007/s00220-019-03487-4
  • Tran, Huy; Yuan, Yizheng A support theorem for SLE curves, Electronic Journal of Probability, Volume 25 (2020) no. none | DOI:10.1214/20-ejp425
  • Gwynne, Ewain; Holden, Nina; Miller, Jason Dimension transformation formula for conformal maps into the complement of an SLE curve, Probability Theory and Related Fields, Volume 176 (2020) no. 1-2, p. 649 | DOI:10.1007/s00440-019-00952-y
  • Gwynne, Ewain; Holden, Nina; Miller, Jason An almost sure KPZ relation for SLE and Brownian motion, The Annals of Probability, Volume 48 (2020) no. 2 | DOI:10.1214/19-aop1385
  • Gwynne, Ewain; Pfeffer, Joshua Connectivity properties of the adjacency graph of SLEκ bubbles for κ(4,8), The Annals of Probability, Volume 48 (2020) no. 3 | DOI:10.1214/19-aop1402
  • Gwynne, Ewain; Miller, Jason; Sheffield, Scott Harmonic functions on mated-CRT maps, Electronic Journal of Probability, Volume 24 (2019) no. none | DOI:10.1214/19-ejp325
  • Miller, Jason; Sheffield, Scott Gaussian free field light cones and SLEκ(ρ), The Annals of Probability, Volume 47 (2019) no. 6 | DOI:10.1214/18-aop1331
  • Miller, Jason; Werner, Wendelin Connection Probabilities for Conformal Loop Ensembles, Communications in Mathematical Physics, Volume 362 (2018) no. 2, p. 415 | DOI:10.1007/s00220-018-3207-8
  • Gwynne, Ewain; Miller, Jason; Sun, Xin Almost sure multifractal spectrum of Schramm–Loewner evolution, Duke Mathematical Journal, Volume 167 (2018) no. 6 | DOI:10.1215/00127094-2017-0049
  • Gwynne, Ewain; Miller, Jason Chordal SLE6 explorations of a quantum disk, Electronic Journal of Probability, Volume 23 (2018) no. none | DOI:10.1214/18-ejp161
  • MILLER, JASON; SHEFFIELD, SCOTT; WERNER, WENDELIN CLE PERCOLATIONS, Forum of Mathematics, Pi, Volume 5 (2017) | DOI:10.1017/fmp.2017.5
  • Flores, S M; Simmons, J J H; Kleban, P; Ziff, R M A formula for crossing probabilities of critical systems inside polygons, Journal of Physics A: Mathematical and Theoretical, Volume 50 (2017) no. 6, p. 064005 | DOI:10.1088/1751-8121/50/6/064005
  • Miller, Jason; Wu, Hao Intersections of SLE Paths: the double and cut point dimension of SLE, Probability Theory and Related Fields, Volume 167 (2017) no. 1-2, p. 45 | DOI:10.1007/s00440-015-0677-x
  • Miller, Jason; Sheffield, Scott Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees, Probability Theory and Related Fields, Volume 169 (2017) no. 3-4, p. 729 | DOI:10.1007/s00440-017-0780-2
  • Zhan, Dapeng Ergodicity of the tip of an SLE curve, Probability Theory and Related Fields, Volume 164 (2016) no. 1-2, p. 333 | DOI:10.1007/s00440-014-0613-5
  • Miller, Jason; Sheffield, Scott Imaginary geometry I: interacting SLEs, Probability Theory and Related Fields, Volume 164 (2016) no. 3-4, p. 553 | DOI:10.1007/s00440-016-0698-0
  • Rohde, Steffen; Zhan, Dapeng Backward SLE and the symmetry of the welding, Probability Theory and Related Fields, Volume 164 (2016) no. 3-4, p. 815 | DOI:10.1007/s00440-015-0620-1
  • Zhan, Dapeng Reversibility of whole-plane SLE, Probability Theory and Related Fields, Volume 161 (2015) no. 3-4, p. 561 | DOI:10.1007/s00440-014-0554-z
  • Sun, Nike Conformally invariant scaling limits in planar critical percolation, Probability Surveys, Volume 8 (2011) no. none | DOI:10.1214/11-ps180

Cité par 23 documents. Sources : Crossref