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October 2020 Square permutations are typically rectangular
Jacopo Borga, Erik Slivken
Ann. Appl. Probab. 30(5): 2196-2233 (October 2020). DOI: 10.1214/19-AAP1555

Abstract

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.

Citation

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Jacopo Borga. Erik Slivken. "Square permutations are typically rectangular." Ann. Appl. Probab. 30 (5) 2196 - 2233, October 2020. https://doi.org/10.1214/19-AAP1555

Information

Received: 1 May 2019; Revised: 1 November 2019; Published: October 2020
First available in Project Euclid: 15 September 2020

MathSciNet: MR4149526
Digital Object Identifier: 10.1214/19-AAP1555

Subjects:
Primary: 05A05, 60C05

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 5 • October 2020
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