  We derive a large deviation principle for random permutations induced by
probability measures of the unit square, called permutons. These permutations
are called $\mu$-random permutations. We also introduce and study a new general
class of models of random permutations, called Gibbs permutation models, which
combines and generalizes $\mu$-random permutations and the celebrated Mallows
model for permutations. Most of our results hold in the general setting of
Gibbs permutation models.
  We apply the tools that we develop to the case of $\mu$-random permutations
conditioned to have an atypical proportion of patterns. Several results are
made more concrete in the specific case of inversions. For instance, we prove
the existence of at least one phase transition for a generalized version of the
Mallows model where the base measure is non-uniform. This is in contrast with
the results of Starr (2009, 2018) on the (standard) Mallows model, where the
absence of phase transition, i.e., phase uniqueness, was proven.
  Our results naturally lead us to investigate a new notion of permutons,
called conditionally constant permutons, which generalizes both
pattern-avoiding and pattern-packing permutons. We describe some properties of
conditionally constant permutons with respect to inversions. The study of
conditionally constant permutons for general patterns seems to be a challenging
problem.
