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

Discrepancy Convergence for the Drunkard's Walk on the Sphere