Limits of permutation sequences through permutation regularity

TL;DR

This paper establishes a framework for understanding the limits of permutation sequences through a measurable function, connecting permutation convergence with a new model of random permutations and their properties.

Contribution

It introduces a limit object for permutation sequences as a Lebesgue measurable function with specific properties, and develops a new model of random permutations.

Findings

Permutation sequences have a natural limit object as a measurable function.

Limit objects are characterized by properties related to distribution functions and mass conditions.

A new model of random permutations generalizes previous models and is of independent interest.

Abstract

A permutation sequence $(\sigma_n)_{n \in \mathbb{N}}$ is said to be convergent if, for every fixed permutation $\tau$, the density of occurrences of $\tau$ in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An important ingredient in the proofs is a new model of random permutations, which generalizes previous models and is interesting for its own sake.