62,999 research outputs found
The scaling limits of planar LERW in finitely connected domains
We define a family of stochastic Loewner evolution-type processes in finitely
connected domains, which are called continuous LERW (loop-erased random walk).
A continuous LERW describes a random curve in a finitely connected domain that
starts from a prime end and ends at a certain target set, which could be an
interior point, or a prime end, or a side arc. It is defined using the usual
chordal Loewner equation with the driving function being plus a
drift term. The distributions of continuous LERW are conformally invariant. A
continuous LERW preserves a family of local martingales, which are composed of
generalized Poisson kernels, normalized by their behaviors near the target set.
These local martingales resemble the discrete martingales preserved by the
corresponding LERW on the discrete approximation of the domain. For all kinds
of targets, if the domain satisfies certain boundary conditions, we use these
martingales to prove that when the mesh of the discrete approximation is small
enough, the continuous LERW and the corresponding discrete LERW can be coupled
together, such that after suitable reparametrization, with probability close to
1, the two curves are uniformly close to each other.Comment: Published in at http://dx.doi.org/10.1214/07-AOP342 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Ergodicity of the tip of an SLE curve
We first prove that, for , a whole-plane
SLE trace stopped at a fixed capacity time satisfies
reversibility. We then use this reversibility result to prove that, for
, a chordal SLE curve stopped at a fixed capacity time
can be mapped conformally to the initial segment of a whole-plane
SLE trace. A similar but weaker result holds for radial
SLE. These results are then used to study the ergodic behavior of an
SLE curve near its tip point at a fixed capacity time. The proofs rely on the
symmetry of backward SLE laminations and conformal removability of SLE
curves for .Comment: 25 pages. Added a remark after Theorem 6.6; added Corollary B.
Loop-Erasure of Plane Brownian Motion
We use the coupling technique to prove that there exists a loop-erasure of a
plane Brownian motion stopped on exiting a simply connected domain, and the
loop-erased curve is the reversal of a radial SLE curve.Comment: 10 page
- …