SLE on doubly-connected domains and the winding of loop-erased random walks

Abstract

Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLE_k with parameter k = 2. In this note, some properties of an SLE_k 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 SLE_2 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 SLE_2 with fixed endpoints. A solution for the endpoint distribution of SLE_4 on the cylinder is obtained and a relation to reflected Brownian motion pointed out.

arXiv comment22 pages, 4 figures

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Cited by in corpus (4)

• Winding statistics of a Brownian particle on a ring
• Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes
• Winding of planar gaussian processes
• A generalisation of Schramm's formula for SLE(2)