2019/04/30 by Jacopo Borga, Erik Slivken · 12 citations
Mathematics · #Advanced Combinatorial Mathematics #Brownian motion #Limit (mathematics) #Limiting #Permutation (music) #Point (geometry) #Point process #Random Matrices and Applications #Random permutation #Square (algebra) #Stochastic processes and statistical mechanics #math.CO #math.PR
paper · pdf · doi:10.1214/19-aap1555
published in The Annals of Applied Probability 30(5) (Institute of Mathematical Statistics) · New version including referee's corrections, accepted for publication in Annals of Applied Probability
openalex created_date 2019/04/11 · arxiv created 2019/12/12 · openalex publication_date 2020/09/15 · arxiv updated 2020/11/10 · openalex updated_date 2026/08/05
We describe the limit (for two topologies) of large uniform random square permutations, that is, 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.