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Lawler, Gregory; Schramm, Oded; Werner, Wendelin

Conformal restriction: The chordal case. (English) Zbl 1030.60096

J. Am. Math. Soc. 16, No. 4, 917-955 (2003).
In previous papers, the authors succeeded in computing the intersection exponents of SLE\(_6\) (stochastic Loewner evolution). The determination of exponents for SLE\(_6\) was also used to compute critical exponents for two-dimensional percolation. In the present paper, the authors go on investigating the chordal restriction measures and their relationship to SLE\(_k\). They prove that there exists a one-parameter family P\(_{\alpha}\) \((\alpha\geq 5/8)\) of such probability measures with P\(_{5/8}\) being exactly the law of chordal SLE\(_{8/3}\). A slightly different restriction property (right-sided property) is also studied; it is related to the reflected Brownian excursion and Bessel-type processes. These results help to understand the scaling limit and exponents of the two-dimensional self-avoiding walk and the relationship between SLE and conformal field theory.
Reviewer: D.Lepingle (Orléans)

Cited in 5 Reviews
Cited in 136 Documents

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B27 Critical phenomena in equilibrium statistical mechanics
60J65 Brownian motion
30C99 Geometric function theory

Keywords:

conformal invariance; restriction property; random fractals; stochastic Loewner evolution
PDFBibTeX XMLCite
\textit{G. Lawler} et al., J. Am. Math. Soc. 16, No. 4, 917--955 (2003; Zbl 1030.60096)
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References:

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