  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.
