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

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
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