Geometry of Permutation Limits

TL;DR

This paper develops a limit theory for permutation processes using permutons, demonstrating that the Archimedean path uniquely minimizes energy and establishing its role as the asymptotic limit for random sorting networks.

Contribution

It introduces a new limit framework for permutation processes and proves the Archimedean path as the unique energy-minimizing limit for random sorting networks.

Findings

The Archimedean path is the unique minimal energy limit.

The Archimedean limit applies to relaxed random sorting networks.

A new limit theory for permutation processes is established.

Abstract

This paper initiates a limit theory of permutation valued processes, building on the recent theory of permutons. We apply this to study the asymptotic behaviour of random sorting networks. We prove that the Archimedean path, the conjectured limit of random sorting networks, is the unique path from the identity to the reverse permuton having minimal energy in an appropriate metric. Together with a recent large deviations result (Kotowski, 2016), it implies the Archimedean limit for the model of relaxed random sorting networks.