  We look at geometric limits of large random non-uniform permutations. We
mainly consider two theories for limits of permutations: permuton limits,
introduced by Hoppen, Kohayakawa, Moreira, Rath, and Sampaio to define a notion
of scaling limits for permutations; and Benjamini-Schramm limits, introduced by
the author to define a notion of local limits for permutations. The models of
random permutations that we consider are mainly constrained models, that is,
uniform permutations belonging to a given subset of the set of all
permutations. We often identify this subset using pattern-avoidance, focusing
on: permutations avoiding a pattern of length three, substitution-closed
classes, (almost) square permutations, permutation families encoded by
generating trees, and Baxter permutations. We explore some universal phenomena
for the models mentioned above. For Benjamini-Schramm limits we explore a
concentration phenomenon for the limiting objects. For permuton limits we
deepen the study of some known universal permutons, called biased Brownian
separable permutons, and we introduce some new ones, called Baxter permuton and
skew Brownian permutons. In addition, for (almost) square permutations, we
investigate the occurrence of a phase transition for the limiting permutons. On
the way, we establish various combinatorial results both for permutations and
other related objects. Among others, we give a complete description of the
feasible region for consecutive patterns as the cycle polytope of a specific
graph; and we find new bijections relating Baxter permutations, bipolar
orientations, walks in cones, and a new family of discrete objects called
coalescent-walk processes.
