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
Highlights
Schramm-Loewner evolution (SLE) is a family of random growth processes invented by O
Schramm conjectured that SLE(2) is the scaling limit of some loop-erased random walks (LERW) and proved his conjuecture with some additional assumptions. He suggested that SLE(6) and SLE(8) should be the scaling limits of certain discrete lattice models
The martingales for annulus SLE8/3 does not help us to prove that annulus SLE8/3 satisfies the restriction property
Summary
Schramm-Loewner evolution (SLE) is a family of random growth processes invented by O. Schramm conjectured that SLE(2) is the scaling limit of some loop-erased random walks (LERW) and proved his conjuecture with some additional assumptions. SLE(8/3) satisfies restriction property, and was conjectured in (8) to be the scaling limit of some self avoiding walk (SAW). Chordal SLE(κ, ρ) processes were invented in (8), and they satisfy one-sided restriction property. If (ξ(t)) = κB(t), 0 ≤ t < p, where κ ≥ 0 and B(t) is a standard linear Brownian motion, Kt and φt, 0 ≤ t < p, are called standard annulus SLEκ hulls and maps, respectively, of modulus p. Since annulus SLE6 is (strongly) equivalent to radial SLE6, so annulus SLE6 satisfies the locality property. Annulus SLE2 is the scaling limit of the corresponding loop-erased random walk. The martingales for annulus SLE8/3 does not help us to prove that annulus SLE8/3 satisfies the restriction property. It seems that annulus SLE8/3 does not satisfy the restriction property
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