  High-dimensional distributions, especially those with heavy tails, are
notoriously difficult for off-the-shelf MCMC samplers: the combination of
unbounded state spaces, diminishing gradient information, and local moves
results in empirically observed ``stickiness'' and poor theoretical mixing
properties -- lack of geometric ergodicity. In this paper, we introduce a new
class of MCMC samplers that map the original high-dimensional problem in
Euclidean space onto a sphere and remedy these notorious mixing problems. In
particular, we develop random-walk Metropolis type algorithms as well as
versions of the Bouncy Particle Sampler that are uniformly ergodic for a large
class of light and heavy-tailed distributions and also empirically exhibit
rapid convergence in high dimensions. In the best scenario, the proposed
samplers can enjoy the ``blessings of dimensionality'' that the convergence is
faster in higher dimensions.
