  We derive the asymptotic winding law of a Brownian particle in the plane
subjected to a tangential drift due to a point vortex. For winding around a
point, the normalized winding angle converges to an inverse Gamma distribution.
For winding around a disk, the angle converges to a distribution given by an
elliptic theta function. For winding in an annulus, the winding angle is
asymptotically Gaussian with a linear drift term. We validate our results with
numerical simulations.
