  We propose a new class of Markov chain Monte Carlo methods, called $k$-polar
slice sampling ($k$-PSS), as a technical tool that interpolates between and
extrapolates beyond uniform and polar slice sampling. By examining Wasserstein
contraction rates and spectral gaps of $k$-PSS, we obtain strong quantitative
results regarding its performance for different kinds of target distributions.
Because $k$-PSS contains uniform and polar slice sampling as special cases, our
results significantly advance the theoretical understanding of both of these
methods. In particular, we prove realistic estimates of the convergence rates
of uniform slice sampling for arbitrary multivariate Gaussian distributions on
the one hand, and near-arbitrary multivariate t-distributions on the other.
Furthermore, our results suggest that for heavy-tailed distributions, polar
slice sampling performs dimension-independently well, whereas uniform slice
sampling suffers a rather strong curse of dimensionality.
