The scaling limits of planar LERW in finitely connected domains

Details The scaling limits of planar LERW in finitely connected domains The scaling limits of planar LERW in finitely connected domains

2006-10-09

arxiv.org/abs/math/0610304v3

Annals of Probability 2008, Vol. 36, No. 2, 467-529

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AOP342 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/07-AOP342

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 $\sqrt{2}B(t)$ 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.

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