High-dimensional permutons: theory and applications

TL;DR

This paper introduces a $d$-dimensional generalization of permutons, extending the theory of permutation limits to higher dimensions and identifying specific high-dimensional permuton limits for classes like Schnyder woods and $d$-separable permutations.

Contribution

The paper develops a new theory of $d$-dimensional permutons and proves convergence of high-dimensional permutations to these limits, with applications to specific permutation classes.

Findings

Identified the 3D permuton limit for Schnyder wood permutations.

Determined the $d$-dimensional permuton limit for $d$-separable permutations.

Connected high-dimensional permutons to objects in random geometry.

Abstract

Permutons, which are probability measures on the unit square $[0, 1]^2$ with uniform marginals, are the natural scaling limits for sequences of (random) permutations. We introduce a $d$-dimensional generalization of these measures for all $d \ge 2$, which we call $d$-dimensional permutons, and extend -- from the two-dimensional setting -- the theory to prove convergence of sequences of (random) $d$-dimensional permutations to (random) $d$-dimensional permutons. Building on this new theory, we determine the random high-dimensional permuton limits for two natural families of high-dimensional permutations. First, we determine the $3$-dimensional permuton limit for Schnyder wood permutations, which bijectively encode planar triangulations decorated by triples of spanning trees known as Schnyder woods. Second, we identify the $d$-dimensional permuton limit for $d$-separable permutations, a pattern-avoiding class of $d$-dimensional permutations generalizing ordinary separable permutations. Both high-dimensional permuton limits are random and connected to previously studied universal 2-dimensional permutons, such as the Brownian separable permutons and the skew Brownian permutons, and share interesting connections with objects arising from random geometry, including the continuum random tree, Schramm--Loewner evolutions, and Liouville quantum gravity surfaces.