## Some properties of annulus SLE.(English)Zbl 1136.82014

The Schramm-Löwner elolution (SLE) is a family of random growth processes investigated by O. Schramm [Isr. J. Math. 118, 221–288 (2000; Zbl 0968.60093)] by connecting Löwner differential equation with a one-dimensional Brownian motion. SLE depends on a single parameter $$\kappa>0$$ and behaves differently for different values of $$\kappa$$. When $$\kappa=2, 8/3, 4 ,6$$ or $$8$$ radial and chordal versions $$\text{SLE}_{\kappa}$$ satisfy some spectral properties. Radial $$\text{SLE}_6$$ satisfies locality property. Since annulus $$\text{SLE}_6$$ is strongly equivalent to radial $$\text{SLE}_6$$, so annulus $$\text{SLE}_6$$ also satisfies the locality property. Annulus $$\text{SLE}_2$$ is the scaling limit of the corresponding loop-erased random walk. The author describes the cases $$\kappa=4,8$$ and $$8/3$$ and finds martingales or local martingales for annulus $$\text{SLE}_{\kappa}$$ in each of these cases. From the local martingale for annulus $$\text{SLE}_{4}$$ he constructs a harmonic explorer whose scaling limit is annulus $$\text{SLE}_{4}$$. The martingales for annulus $$\text{SLE}_{8/3}$$ are similar to the martingales for radial and chordal $$\text{SLE}_{8/3}$$, which are used to show that radial and chordal $$\text{SLE}_{8/3}$$ satisfy the restriction property. However, the martingales for annulus $$\text{SLE}_{8/3}$$ does not help to prove that annulus $$\text{SLE}_{8/3}$$ satisfies the restriction property. On the contrary, it seems that annulus $$\text{SLE}_{8/3}$$ does not satisfy the restriction property.

### MSC:

 82B27 Critical phenomena in equilibrium statistical mechanics 82B43 Percolation 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60D05 Geometric probability and stochastic geometry 30C35 General theory of conformal mappings

Zbl 0968.60093
Full Text: