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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

Citations:

Zbl 0968.60093
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