The permuton limit of strong-Baxter and semi-Baxter permutations is the skew Brownian permuton

TL;DR

This paper proves that strong-Baxter and semi-Baxter permutations converge to a new universal permuton called the skew Brownian permuton, using a novel encoding technique with multi-dimensional walks.

Contribution

It introduces a new convergence result for specific permutation classes towards the skew Brownian permuton and develops a robust encoding technique for such proofs.

Findings

Strong-Baxter permutations converge to the skew Brownian permuton.

Semi-Baxter permutations also converge to the skew Brownian permuton.

A new technique using generating trees and multi-dimensional walks is developed.

Abstract

We recently introduced a new universal family of permutons, depending on two parameters, called skew Brownian permuton. For some specific choices of the parameters, the skew Brownian permuton coincides with some previously studied permutons: the biased Brownian separable permuton and the Baxter permuton. The latter two permutons are degenerate cases of the skew Brownian permuton. In the present paper we prove the first convergence result towards a non-degenerate skew Brownian permuton. Specifically, we prove that strong-Baxter permutations converge in the permuton sense to the skew Brownian permuton for a non-degenerate choice of the two parameters. In order to do that, we develop a robust technique to prove convergence towards the skew Brownian permuton for various families of random constrained permutations. This technique relies on generating trees for permutations, allowing an encoding of permutations with multi-dimensional walks in cones. We apply this technique also to semi-Baxter permutations.