  Mallows permutation model, introduced by Mallows in statistical ranking
theory, is a class of non-uniform probability measures on the symmetric group
$S_n$. The model depends on a distance metric $d(\sigma,\tau)$ on $S_n$, which
can be chosen from a host of metrics on permutations. In this paper, we focus
on Mallows permutation models with $L^1$ and $L^2$ distances, respectively
known in the statistics literature as Spearman's footrule and Spearman's rank
correlation.
  Unlike most of the random permutation models that have been analyzed in the
literature, Mallows permutation models with $L^1$ and $L^2$ distances do not
have an explicit expression for their normalizing constants. This poses
challenges to the task of sampling from these Mallows models. In this paper, we
consider hit and run algorithms for sampling from both models. Hit and run
algorithms are a unifying class of Markov chain Monte Carlo (MCMC) algorithms
including the celebrated Swendsen-Wang and data augmentation algorithms. For
both models, we show order $\log{n}$ mixing time upper bounds for the hit and
run algorithms. This demonstrates much faster mixing of the hit and run
algorithms compared to local MCMC algorithms such as the Metropolis algorithm.
The proof of the results on mixing times is based on the path coupling
technique, for which a novel coupling for permutations with one-sided
restrictions is involved.
  Extensions of the hit and run algorithms to weighted versions of the above
models, a two-parameter permutation model that involves the $L^1$ distance and
Cayley distance, and lattice permutation models in dimensions greater than or
equal to $2$ are also discussed. The order $\log{n}$ mixing time upper bound
pertains to the two-parameter permutation model.
