  We analyze the drunkard's walk on the unit sphere with step size theta and
show that the walk converges in order constant/sin^2(theta) steps in the
discrepancy metric. This is an application of techniques we develop for
bounding the discrepancy of random walks on Gelfand pairs generated by
bi-invariant measures. In such cases, Fourier analysis on the acting group
admits tractable computations involving spherical functions. We advocate the
use of discrepancy as a metric on probabilities for state spaces with isometric
group actions.
