Baxter permuton and Liouville quantum gravity

TL;DR

This paper derives an explicit formula for the Baxter permuton's density, proves all pattern densities are positive, and connects these findings to Liouville quantum gravity and SLE, introducing the skew Brownian permuton.

Contribution

It provides an explicit density formula for the Baxter permuton and extends the framework to a new family called skew Brownian permuton using LQG and SLE.

Findings

Explicit formula for Baxter permuton's density.

All pattern densities of Baxter permuton are positive.

Expected inversions proportion relates to LQG/SLE parameters.

Abstract

The Baxter permuton is a random probability measure on the unit square which describes the scaling limit of uniform Baxter permutations. We find an explict formula for the expectation of the Baxter permuton, i.e.\ the density of its intensity measure. This answers a question of Dokos and Pak (2014). We also prove that all pattern densities of the Baxter permuton are strictly positive, distinguishing it from other permutons arising as scaling limits of pattern-avoiding permutations. Our proofs rely on a recent connection between the Baxter permuton and Liouville quantum gravity (LQG) coupled with the Schramm-Loewner evolution (SLE). The method works equally well for a two-parameter generalization of the Baxter permuton recently introduced by the first author, except that the density is not as explicit. This new family of permutons, called \emph{skew Brownian permuton}, describes the scaling limit of a number of random constrained permutations. We finally observe that in the LQG/SLE framework, the expected proportion of inversions in a skew Brownian permuton equals $\frac{\pi-2\theta}{2\pi}$ where $\theta$ is the so-called imaginary geometry angle between a certain pair of SLE curves.