We describe the limit (for two topologies) of large uniform random square permutations, *i.e.*, permutations where every point is a record. First we describe the global behavior by showing 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. This is a joint work with E.Slivken

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