SLE on Doubly-Connected Domains and the Winding of Loop-Erased Random Walks


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Abstract

Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLEκ with parameter κ=2. In this note, some properties of an SLEκ trace on doubly-connected domains are studied and a connection to passive scalar diffusion in a Burgers flow is emphasised. In particular, the endpoint probability distribution and winding probabilities for SLE2 on a cylinder, starting from one boundary component and stopped when hitting the other, are found. A relation of the result to conditioned one-dimensional Brownian motion is pointed out. Moreover, this result permits to study the statistics of the winding number for SLE2 with fixed endpoints. A solution for the endpoint distribution of SLE4 on the cylinder is obtained and a relation to reflected Brownian motion pointed out.

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