Large deviations for the interchange process on the interval and incompressible flows

TL;DR

This paper establishes a connection between large deviations in the interchange process and incompressible flows, using permuton processes and Dirichlet energy, with implications for counting relaxed sorting networks.

Contribution

It introduces a rigorous framework linking permutation processes to incompressible Euler equations via large deviations and Dirichlet energy, extending understanding of sorting networks.

Findings

Large deviations are controlled by Dirichlet energy.

Connection established between permutation processes and Euler equations.

Asymptotic counting of relaxed sorting networks achieved.

Abstract

We use the framework of permuton processes to show that large deviations of the interchange process are controlled by the Dirichlet energy. This establishes a rigorous connection between processes of permutations and one-dimensional incompressible Euler equations. While our large deviation upper bound is valid in general, the lower bound applies to processes corresponding to incompressible flows, studied in this context by Brenier. These results imply the Archimedean limit for relaxed sorting networks and allow us to asymptotically count such networks.