  For a positive number $q$ the Mallows measure on the symmetric group is the
probability measure on $S_n$ such that $P_{n,q}(\pi)$ is proportional to
$q$-to-the-power-$\mathrm{inv}(\pi)$ where $\mathrm{inv}(\pi)$ equals the
number of inversions: $\mathrm{inv}(\pi)$ equals the number of pairs $i<j$ such
that $\pi_i>\pi_j$. One may consider this as a mean-field model from
statistical mechanics. The weak large deviation principle may replace the Gibbs
variational principle for characterizing equilibrium measures. In this sense,
we prove absence of phase transition, i.e., phase uniqueness.
