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

TL;DR

This paper introduces coalescent-walk processes and establishes their connection to Baxter permutations and related structures, proving their convergence to a new fractal permuton and analyzing the scaling limits of these combinatorial objects.

Contribution

It introduces coalescent-walk processes and demonstrates their role in connecting Baxter permutations, plane bipolar orientations, and walks, including establishing their scaling limits and convergence to a new fractal permuton.

Findings

Joint Benjamini--Schramm convergence of the four families.

Construction of the coalescent Baxter permuton as a scaling limit.

Convergence of coalescent-walk processes to a perturbed Tanaka SDE.

Abstract

Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant 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, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections. 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 fractal random measure of the unit square, called the coalescent Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a continuous random coalescent-walk process encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections (to be explored in future projects) with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations. We further prove some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case.