## Electronic Journal of Probability

### From CLE($\kappa$) to SLE($\kappa,\rho$)'s

#### Abstract

We show how to connect together the loops of a simple Conformal Loop Ensemble (CLE) in order to construct samples of chordal SLE$_{\kappa}$ processes and their SLE$_{\kappa}(\rho)$ variants, and we discuss some consequences of this construction.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 36, 20 pp.

Dates
Accepted: 12 March 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064261

Digital Object Identifier
doi:10.1214/EJP.v18-2376

Mathematical Reviews number (MathSciNet)
MR3035764

Zentralblatt MATH identifier
1338.60205

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE)

Rights

#### Citation

Werner, Wendelin; Wu, Hao. From CLE($\kappa$) to SLE($\kappa,\rho$)'s. Electron. J. Probab. 18 (2013), paper no. 36, 20 pp. doi:10.1214/EJP.v18-2376. https://projecteuclid.org/euclid.ejp/1465064261

#### References

• Dmitry Chelkak, Hugo Duminil-Copin, Clément Hongler, Antti Kemppainen and Stanislav Smirnov. Convergence of Ising interfaces to Schramm's SLEs. preliminary version, 2012
• Chelkak, Dmitry; Smirnov, Stanislav. Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189 (2012), no. 3, 515–580.
• Dubédat, Julien. ${\rm SLE}(\kappa,\rho)$ martingales and duality. Ann. Probab. 33 (2005), no. 1, 223–243.
• Dubédat, Julien. Duality of Schramm-Loewner evolutions. Ann. Sci. Ã‰c. Norm. Supér. (4) 42 (2009), no. 5, 697–724.
• Lawler, Gregory; Schramm, Oded; Werner, Wendelin. Conformal restriction: the chordal case. J. Amer. Math. Soc. 16 (2003), no. 4, 917–955 (electronic).
• Lawler, Gregory F. Conformally invariant processes in the plane. Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, RI, 2005. xii+242 pp. ISBN: 0-8218-3677-3
• Lawler, Gregory F.; Werner, Wendelin. The Brownian loop soup. Probab. Theory Related Fields 128 (2004), no. 4, 565–588.
• Mörters, Peter; Peres, Yuval. Brownian motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. xii+403 pp. ISBN: 978-0-521-76018-8
• Jason Miller and Scott Sheffield. Imaginary geometry i: Interacting SLEs. Preprint, 2012.
• Jason Miller and Scott Sheffield. Imaginary geometry ii: reversibility of $\mathrm{SLE}_{\kappa}(\rho_1;\rho_2)$ for $\kappa \in (0,4)$. Preprint, 2012.
• Jason Miller and Hao Wu. Intersections of SLE Paths: the double and cut point dimension of SLE. Preprint, 2013.
• Nacu, Şerban; Werner, Wendelin. Random soups, carpets and fractal dimensions. J. Lond. Math. Soc. (2) 83 (2011), no. 3, 789–809.
• Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7
• Schramm, Oded. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221–288.
• Schramm, Oded; Sheffield, Scott; Wilson, David B. Conformal radii for conformal loop ensembles. Comm. Math. Phys. 288 (2009), no. 1, 43–53.
• Schramm, Oded; Wilson, David B. SLE coordinate changes. New York J. Math. 11 (2005), 659–669 (electronic).
• Schramm, Oded; Zhou, Wang. Boundary proximity of SLE. Probab. Theory Related Fields 146 (2010), no. 3-4, 435–450.
• Sheffield, Scott. Exploration trees and conformal loop ensembles. Duke Math. J. 147 (2009), no. 1, 79–129.
• Sheffield, Scott; Werner, Wendelin. Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. of Math. (2) 176 (2012), no. 3, 1827–1917.
• Werner, Wendelin. SLEs as boundaries of clusters of Brownian loops. C. R. Math. Acad. Sci. Paris 337 (2003), no. 7, 481–486.
• Werner, Wendelin. Girsanov's transformation for ${\rm SLE}(\kappa,\rho)$ processes, intersection exponents and hiding exponents. Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 1, 121–147.
• Werner, Wendelin. Conformal restriction and related questions. Probab. Surv. 2 (2005), 145–190.
• Wendelin Werner and Hao Wu. On conformally invariant CLE explorations. Comm. Math. Phys., to appear.
• Zhan, Dapeng. Reversibility of chordal SLE. Ann. Probab. 36 (2008), no. 4, 1472–1494.
• Zhan, Dapeng. Reversibility of some chordal ${\rm SLE}(\kappa;\rho)$ traces. J. Stat. Phys. 139 (2010), no. 6, 1013–1032.