Square permutations are typically rectangular (Q2657936)

This paper studies limits of random permutations that are sampled from the class of all permutations in which all elements are records, that is, every element is either a maximum or a minimum from the left or right. These are called square permutations. The first result is about the limiting object of random permutations whose anchored pairs are sampled uniformly at random. Roughly speaking, the latter are objects that describe the shape of a permutation. In particular, they identify a certain permuton which is the limit. Furthermore, the authors give results about the fluctuations around the limiting permuton. The limiting local topology of permutations is also considered, which is the analogue of the Benjamini-Schramm convergence. In this setting, there is an appropriate definition of the neighbourhood of an element within a permutation. Here, the (bounded) neighbourhood of a random element of a uniformly random square permutation is considered and a (random) limiting object is determined. Finally, the limiting proportions of the number of occurrences within a random square permutation of a fixed pattern are considered. In particular, examples of non-concentration are provided in this setting.