# Square permutations are typically rectangular

## TL;DR

This paper studies the asymptotic structure of large uniform square permutations, revealing their global limit as a random rectangle, local behavior, and pattern occurrence frequencies, with connections to Brownian motion.

## Contribution

It introduces a sampling procedure for uniform square permutations and characterizes their global and local limits, including pattern frequencies and fluctuations.

## Key findings

- Permuton limit described by a random rectangle
- Fluctuations characterized by coupled Brownian motions
- Asymptotic pattern occurrence proportions determined

## Abstract

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.