  Introduced by Mallows as a ranking model in statistics, Mallows permutation
model is a class of non-uniform probability distributions on the symmetric
group $S_n$. The model depends on a distance metric on $S_n$ and a scale
parameter $\beta$. In this paper, we take the distance metric to be the $L^1$
distance (also known as Spearman's footrule in the statistics literature), and
investigate the cycle structure of random permutations drawn from Mallows
permutation model with the $L^1$ distance.
  We focus on the parameter regime where $\beta>0$. We show that the expected
length of the cycle containing a given point is of order
$\min\{\max\{\beta^{-2},1\},n\}$, and the expected diameter of the cycle
containing a given point is of order $\min\{e^{-2\beta}\max\{\beta^{-2},1\},
n-1\}$. Moreover, when $\beta\ll n^{-1\slash 2}$, the sorted cycle lengths (in
descending order) normalized by $n$ converge in distribution to the
Poisson-Dirichlet law with parameter $1$. The proofs of the results rely on the
hit and run algorithm, a Markov chain for sampling from the model.
