SLE intersecting with random hulls

Yong Han; Yuefei Wang; Michel Zinsmeister

doi:10.1016/j.crma.2019.04.005

Probability theory

SLE intersecting with random hulls
[Intersection de SLE avec une enveloppe aléatoire]

Présenté par : the Editorial Board

Yong Han ¹ ; Yuefei Wang ² ; Michel Zinsmeister ³

¹ Yau Mathematical Sciences Center, Tsinghua University, Beijng, 100084, China
² Institute of Mathematics, Academy of Mathematics and Systems Sciences and University of the Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China
³ IDP, Université d'Orléans, BP 6759, 45067 Orléans cedex 2, France

Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 395-400.

Résumé
Abstract

Lawler, Schramm et Werner ont donné en 2003 une formule explicite pour la probabilité que $SLE (8 / 3)$ ne rencontre pas une enveloppe déterministe. Pour $SLE (κ)$ avec $κ \neq 8 / 3$ , aucune formule de ce type ne semble connue. Nous considérons ici une enveloppe aléatoire engendrée par une mesure de restriction conforme indépendante et nous obtenons une formule explicite de la probabilité que $SLE (κ)$ ne la rencontre pas lorsque $κ \in (0, 8)$ . Comme corollaire, nous donnons une nouvelle démonstration d'un résultat de Werner sur les mesures de restrictions conformes.

Lawler, Schramm, and Werner gave in 2003 an explicit formula of the probability that $SLE (8 / 3)$ does not intersect a deterministic hull. For general $SLE (κ)$ with $κ \neq 8 / 3$ , no such explicit formula has been obtained so far. In this paper, we shall consider a random hull generated by an independent chordal conformal restriction measure and obtain an explicit formula for the probability that $SLE (κ)$ does not intersect this random hull for any $κ \in (0, 8)$ . As a corollary, we will give a new proof of Werner's result on conformal restriction measures.

Reçu le : 2018-08-11
Accepté le : 2019-04-12
Publié le : 2019-04-01

DOI : 10.1016/j.crma.2019.04.005

Affiliations des auteurs :

Yong Han ¹ ; Yuefei Wang ² ; Michel Zinsmeister ³

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     author = {Yong Han and Yuefei Wang and Michel Zinsmeister},
     title = {SLE intersecting with random hulls},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {395--400},
     publisher = {Elsevier},
     volume = {357},
     number = {4},
     year = {2019},
     doi = {10.1016/j.crma.2019.04.005},
     language = {en},
}

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Yong Han; Yuefei Wang; Michel Zinsmeister. SLE intersecting with random hulls. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 395-400. doi : 10.1016/j.crma.2019.04.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.04.005/

Bibliographie
Cité par

[1] V. Beffara The dimension of the SLE curves, Ann. Probab., Volume 36 (2002), pp. 1421-1452

[2] G. Lawler; O. Schramm Conformal invariance of planar loop-erased random walks and uniform spanning trees, Selected Works of Oded Schramm, Springer, 2011, pp. 931-987

[3] G. Lawler; O. Schramm; W. Werner Conformal restriction: the chordal case, J. Amer. Math. Soc., Volume 16 (2003), pp. 917-955

[4] G.F. Lawler Conformally Invariant Processes in the Plane, vol. 114, American Mathematical Society, Providence, RI, USA, 2008

[5] G.F. Lawler; O. Schramm; W. Werner Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math., Volume 187 (2001), pp. 237-273

[6] G.F. Lawler; O. Schramm; W. Werner Values of Brownian intersection exponents. II. Plane exponents, Acta Math., Volume 187 (2001), pp. 275-308

[7] G.F. Lawler; O. Schramm; W. Werner On the scaling limit of planar self-avoiding walk, Mathematics, Volume 2 (2002), pp. 339-364

[8] E. Peltola; H. Wu Global Multiple SLEs for $κ \leq 4$ and Connection Probabilities for Level Lines of GFF, 2017

[9] S. Rohde; O. Schramm Basic properties of SLE, Selected Works of Oded Schramm, Springer, 2011, pp. 989-1030

[10] O. Schramm Scaling limits of loop-erased random walks and uniform spanning trees, Isr. J. Math., Volume 118 (2000), pp. 221-288

[11] H. Wu Conformal restriction and brownian motion, Probab. Surv., Volume 12 (2014)

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