  We theoretically analyze the properties of a geodesic random walk on the
Euclidean $d$-sphere. Specifically, we prove that the random walk's transition
kernel is Wasserstein contractive with a contraction rate which can be bounded
from above independently of the dimension $d$. Our result is of particular
interest due to its implications regarding the potential for
dimension-independent performance of both geodesic slice sampling on the sphere
and Gibbsian polar slice sampling, which are Markov chain Monte Carlo methods
for approximate sampling from essentially arbitrary distributions on their
respective state spaces.
