Electronic Journal of Probability

Some Properties of Annulus SLE

Dapeng Zhan

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An annulus $\mathrm{SLE}_{\kappa}$ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus $\mathrm{SLE}_4$, $\mathrm{SLE}_{8}$ and $\mathrm{SLE}_{8/3}$. From the local martingale for annulus $\mathrm{SLE}_4$ we find a candidate of discrete lattice model that may have annulus $\mathrm{SLE}_4$ as its scaling limit. The local martingale for annulus $\mathrm{SLE}_{8/3}$ is similar to those for chordal and radial $\mathrm{SLE}_{8/3}$. But it seems that annulus $\mathrm{SLE}_{8/3}$ does not satisfy the restriction property

Article information

Electron. J. Probab., Volume 11 (2006), paper no. 41, 1069-1093.

Accepted: 28 November 2006
First available in Project Euclid: 31 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B27: Critical phenomena
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 30C35: General theory of conformal mappings

continuum scaling limit percolation SLE conformal invariance

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Zhan, Dapeng. Some Properties of Annulus SLE. Electron. J. Probab. 11 (2006), paper no. 41, 1069--1093. doi:10.1214/EJP.v11-338. https://projecteuclid.org/euclid.ejp/1464730576

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