Electronic Journal of Probability

Some Properties of Annulus SLE

Dapeng Zhan

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730576

Digital Object Identifier
doi:10.1214/EJP.v11-338

Mathematical Reviews number (MathSciNet)
MR2268538

Zentralblatt MATH identifier
1136.82014

Subjects
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

Keywords
continuum scaling limit percolation SLE conformal invariance

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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