On the approximation of permutons

TL;DR

This paper investigates how well permutons can be approximated by finite permutations, revealing limitations based on the structure of the underlying measure-preserving functions.

Contribution

It transfers discrepancy bounds to permutation approximation, identifies conditions for superlinear approximation, and analyzes the approximation limits of specific permutons.

Findings

Superlinear approximation occurs only for permutons supported by measure-preserving graphs.

Local regularity of the measure-preserving function hinders approximability.

Lower bounds on approximation error are established for the biased Brownian separable permuton.

Abstract

We study the optimal rectangular-discrepancy approximation of permutons by finite permutations. We transfer bounds from discrepancy theory to this more restricted setup. Moreover, we show that superlinear approximation can occur only for permutons supported by graphs of measure-preserving functions, and demonstrate how the local regularity of this function obstructs approximability. We also consider the biased Brownian separable permuton and prove a lower bound on its approximation error by showing that its supporting measure-preserving function has Lipschitz points almost surely.