  We describe the limit (for two topologies) of large uniform random square
permutations, i.e., permutations where every point is a record. The starting
point for all our results is a sampling procedure for asymptotically uniform
square permutations. Building on that, we first describe the global behavior by
showing that these permutations have a permuton limit which can be described by
a random rectangle. We also explore fluctuations about this random rectangle,
which we can describe through coupled Brownian motions. Second, we consider the
limiting behavior of the neighborhood of a point in the permutation through
local limits. As a byproduct, we also determine the random limits of the
proportion of occurrences (and consecutive occurrences) of any given pattern in
a uniform random square permutation.
