Recently, Dong and Liechty determined the large-n asymptotic behavior of n Brownian walkers on the unit circle with non-crossing paths conditioned to start from a single point at time zero and end at the same point at a fixed ending time. We analyze the analogous problem with a nonzero drift. We show there is a critical drift value for which the total winding is asymptotically zero with probability one. We compute the critical drift explicitly and discuss the positive winding case. Our results follow from asymptotic analysis of related discrete orthogonal polynomials carried out via the nonlinear steepest-descent method for Riemann-Hilbert problems. This is joint work with Karl Liechty. Speaker(s): Robert Buckingham (University of Cincinnati)