Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

TL;DR

This paper introduces coalescent-walk processes related to Baxter permutations and bipolar orientations, proves their joint convergence, constructs a new Baxter permuton as a scaling limit, and relates these objects through local and global limits.

Contribution

It introduces coalescent-walk processes, establishes their convergence, and constructs the Baxter permuton as a new scaling limit for Baxter permutations.

Findings

Joint Benjamini--Schramm convergence for four families.

Construction of the Baxter permuton as a new limit object.

Extension of previous results to dual tandem walks and local limits.

Abstract

Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant, called tandem walks, are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes and we relate it to the three families mentioned above. We prove joint Benjamini--Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new random measure on the unit square, called the Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. In addition, we relate the limiting objects of the four families to each other, both in the local and scaling limit case. The scaling limit result is based on the convergence of the trajectories of the coalescent-walk process to the coalescing flow -- in the terminology of Le Jan and Raimond (2004) -- of a perturbed version of the Tanaka stochastic differential equation. Our scaling result entails joint convergence of the tandem walks of a plane bipolar orientation and its dual, extending the main result of Gwynne, Holden, Sun (2016), and giving an alternative answer to Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (2019) compared to the one of Gwynne, Holden, Sun (2016).